About the Rules
You are correct that even if you can do something as much as you want, it still takes time to do. Something that takes your action in combat should not be do-able more frequently than once per 6 seconds outside of combat most of the time (there can be some exceptions, but spellcasting would not usually be considered a candidate for that). So far, so good.
And you are also correct to point out that spamming fire bolt is no different from spamming “club.” For all we know, it’s actually easier—we know that swinging a club takes a certain amount of physical effort, but we have no real-world experience with cantrips and so don’t really know how much effort that would take.
But D&D 5e doesn’t include rules for getting tired faster when you’re working hard: you get tired when you are forced to be tired by magic or some other effect. Even going without sleep doesn’t apply any penalties—you have to make a Constitution check to do it, but if you pass that check, you’re right as rain without sleep. Regular activities do not cause fatigue in D&D 5e.
Likewise, there aren’t rules in D&D 5e for your club wearing down as you bash things with it. Damage to items happens from particular effects that damage items, not through regular use.
And the reason why is because D&D 5e is trying to emulate a very particular form of high-adventure, high-fantasy narrative. The focus is on the quest, the defeat of enemies, the thwarting of schemes, the saving of the hostage/town/kingdom/world. Its rules are focused on bringing those things to life—and it tends to avoid rules that aren’t crucial to doing so.
That’s because rules have costs. They cost Wizards of the Coast, in that people have to spend time thinking up, writing down, editing, and testing those rules, which is significant because time is money and they’re in the business of making money, but also because players want more material to play with and time they spend on extraneous details means less time for the big stuff.
And even if you somehow magically got the rules without it cutting into Wizards’ time to produce other content, rules have costs at the table, too. You have to read them, understand them, remember them, and adjudicate them at the table. You have to keep track of more details, you have to apply more temporary effects to your character sheet, and you have to deal with more corner cases.
About What D&D Means to You, Means to Your Player, Means to Wizards of the Coast
All of this is to say, you and your friend have different ideas about what kind of game you’re trying to play—and D&D 5e certainly seems to be a bit more on your friend’s side than yours. D&D is a game of fantastical characters doing fantastical things—overcoming human limitations is entirely par for the course. And I mean, people in construction or demolition may very well spend something like eight hours a day swinging an axe or hammer to break things down (or put things together), so it’s not as if this really is superhuman.
There are “grittier” game systems that track these kinds of things with more detail, that apply penalties to characters as they perform regular tasks without breaks (whether those are rest for themselves or care for their equipment). D&D 5e does not; it ignores or abstracts those away. That is because of the kind of game D&D 5e is trying to be. You can change D&D 5e to be more of the kind of game you want—but be aware that this may not be a change your group wants. This player, for instance, seems quite happy with his “fire bolt it ’til it’s gone” strategy. He or she may not like a game where you have to spend time and effort tracking your activities and applying penalties for fatigue, penalties that discourage strategies like this. That may very well sound like un-fun work to this player, whose result is a negative thing that inhibits their enjoyment of the game. At the extreme end, a change like that could make the game “no longer D&D” for them, and they may not feel like they’re getting what they signed up for—and they may not want to continue playing. So changes you make should be done with care and consideration for the group as a whole, and deserve a discussion with the group. D&D gives the DM the authority to make these kinds of changes, but D&D doesn’t give the DM the authority to make a player play, so it behooves you to avoid making sweeping changes like this by fiat alone.
So the problem that you actually have is that the player is trying to turn to fire bolt as their sole solution to all problems, which makes for a repetitive, unfun game where the overwhelming majority of the actual system (to say nothing of the other players) don’t actually come into play. One reason someone might not do that is if they got tired, hence this question—but you might be falling into the X–Y trap here, because that isn’t the only reason why someone might not see fire bolt as their one-stop shopping for solutions.
An obvious other reason to not use fire bolt is time; fire bolt doesn’t do a whole lot of damage so it’s going to take an awfully long time to get through anything big and sturdy with it. And D&D characters are frequently pressed for time, so that’s definitely a good issue to raise—but D&D characters are not always pressed for time, and some D&D characters may not be very often at all, if they are more adventurers and treasure hunters than world savers. Clearly, since this strategy has worked for them in the past, your players have not always been pressed for time.
But the really big reason why fire bolt isn’t usually seen as a versatile multitasker is because it really only ever does one thing: a small amount of fire damage. Think of the real world: do you see flamethrowers as a great, common solution to a lot of problems? I don’t; we don’t even use them for warfare anymore because flamethrowers were particularly suited to trenches and small bunkers and we don’t see a lot of those any more. So even as a weapon, they’re a pretty niche weapon.
So when you say that fire bolt has become an overly-centralizing tactic in your games, I start to suspect that you need more variety in the challenges presented to your players. I mean, it’s awfully hard to see how fire bolt helps in any kind of social situation (I guess you could threaten someone with it, but threats are often not a great tool, and even then you still need the Charisma to make the threat convincing). For that matter, fire bolt is often not a great choice for actual combat, because you want to deal more damage than that (enemies aren’t going to just sit there and let you plink away at their hp, they’re going to trying to do something about your hp).
Need to make the players work to get through somewhere? Consider a chasm—can’t rightly see how a fire bolt is going to help you cross one of those. Magical fire-proof wards are another thought, though a bit lame—feel free to break those out some of the time, but relying on them too much will come off as punishing the player. Or, make something on the other side highly flammable! So ok, sure, you managed to melt a hole through the steel door... and the heat of the door and/or some stray sparks from the fire bolt set fire to the priceless shroud you were trying to get at. Or, worse, the barrels of explosives stuck in there with the treasure. (Try to give the players warning about this kind of thing, though, or again it can seem like petty revenge for overuse of fire bolt, which is un-fun.)
Ideas for player challenges abound, and it would be way too much for an answer here to really try to get into all of them. But I do want to leave you with one more thing: The Angry DM’s 5 Simple Rules for Dating My Teenaged Skill System. The Angry DM is highly opinionated and, well, angry, and uses a lot of (self-censored) swearing in his blog posts, so fair warning there, and I disagree with him strongly on a number of things—but on this I think he’s spot-on. He’s talking about D&D 4e in his blog, so some of the specific mechanics work a little differently, but what you’re really interested in are the ideas.
In particular, he addresses the idea of players doing the same thing repeatedly until it works: he treats that as entirely predictable and expected. He talks about this in “Rule #2: Only Roll When There is Chance of Success, A Chance of Failure, and A Risk or Cost of Failure,” so that’s in the context of whether or not you should ask for a die roll—but it’s also relevant to whether or not you should be concerned about this behavior in the first place.
The assumption is that, lacking any constraints, the party will keep trying something over and over until they succeed. The DM should take that into account. When the party attempts an action, assume they mean to keep trying until it succeeds. If the party could freely do so, then it is not worth rolling. They succeed. And beware not to impose constraints that don’t really exist. “Because it will take an hour” is not a constraint. “Because it will take an hour and the place will explode in two hours” is a constraint.
Here, he suggests embracing this kind of behavior, expecting it, building it into your plans. I suggest that as well. You don’t need a fiddly (or fiat-y) fatigue system to prevent “fire bolt it ’til it’s gone” from becoming the be-all, end-all solution to problems. You just need to plan for it.