Stacking wins by numbers, but you can do both
My main recommendation is to not focus to myopically on maximizing skill rolls. You also can mix-and-match and benefit from both expertise and stacking bonuses as relatively little cost, so you may not need to make an either-or choice on it.
What is numerically better?
As an optimization question for maximizing bonuses to a any given skill roll, this comes down to how you can build a character that has abilities to grant bonuses, and what the expected contribution of those is. You can then compare this vs the character development investment needed to pick up more skills for the flat bonus from having Expertise in the skill. I'll assume as you are talking about the build of your character for this, no help from others for the comparison. Here are a few stackable bonuses, along with some sources to get them. This is focused on skill bonuses, not saves, as you say you are looking to build a skill monkey:
Fortune of the Many: +3 (number of allies) from Hobgolin race; usable proficiency bonus times per long rest
Guidance cantrip: +1d4 (2.5) - Cleric 1, or spell feats that give cantrips like Magic Initiate
Emboldening Bond: +d4 (2.5) to a save/check - Peace Domain cleric 1, limited to proficiency bonus times per long rest
Dark One's Own Luck: +d10 (5.5) to save/check - Fiend warlock 6, once per short rest
Flash of Genius: + Int mod, if optimized +4 to +5 (outside of Tome shenanigans) - Artificer 7, limited to Int mod times per long rest
I left off Bardic Inspiration, as that does not work on yourself, only on others, and magic items, as these would help both build approaches.
You can pick up every permanent skill by level 7, if you optimize for that, 6 of them with expertise, with a bonus of +3 or +6. From first level, the ability stacking approach with a Hobgoblin Peace Domain cleric 1 would already give you +8, so going for skills just based on the numbers is not worthwhile.
Also in the endgame by level 14, with a Hobgoblin Peace Domain Cleric 1/Warlock 6/Artificer 7 you would end up with +3 + 2.5 + 2.5 + 5.5 + 5 = +18.5 total bonus. The total bonus from Expertise at level 14 is +10. So, numerically stacking bonuses will get the higher number by far, just as RPGbot claims, and it would not be a worthwile strategy trying to maximize picking up as many skills as possible. (And that is before considering bonuses may work for tool proficiencies and saves, too).
In practice, I think getting the highest bonus on all skills with stacking is not worthwhile to pursue too narrowly. Proficiency or expertise may work out better in several ways, even if the numerical bonus is smaller:
First, many of these bonus-stacking shenanigans are limited to a number of times per day. If you need lots of investigation checks and perception checks, and sneak around with stealth checks, you will run out of applications for stacking other than guidance. Expertise does not have that problem, it is an unlimited resource.
Second, there are situations where it would be awkward to stack these bonuses. For example in social encounters - you are at a noble's reception and want to impress your opposite number with your witty banter. Are you going to cast a spell (which is perceptible) in front of them asking your deity for guidance in your next quip?
Third, casting a spell like guidance has an action cost. If you need to use your skill in combat, that is going to be an issue. Expertise does not have that downside.
On the other hand, being able to add bonuses may be more useful than expertise, too. Many checks do require all in the party to succeed, or at least have a successful party check. You all need to climb up that wall. You all need to sneak by the guard. You all need to swim savely to the shore. In such situations, having a single "skill moneky" character will not suffice, and helping out the weaker characters is of value.
Note however that the cleric which contributes a big part to stacking only needs a single level of investment. So there is no need to choose one or the other - you can have both. You have unused levels, and throwing in a level of cleric with one or three of bard, and a level of rogue is entirely feasible. Because what is better than bounded-accuracy breaking expertise or bounded-accuracy breaking bonus stacking from power-creep subclasses and races? Bounded-accuracy breaking expertise and bounded-accuracy breaking bonus stacking from power-creep subclasses and races -- at least if that is your kind of thing.
From your earlier posts, your character is already set as a Bard, and your race is already set, too. The cheapest option, if you don't care for Bard spell progression and if your DM has not banned it like many have, may be to pick up a single level in the broken Peace Domain cleric, and get a boost of +2.5 to +5 on all your checks, which is going to be as good as expertise. You really don't need higher levels in that class for skill boosting. You still can continue to level 3 bard for the expertise (and if you pick lore, 3 more skills).
Depending what you like you could then get 6 levels in warlock or 7 in artificer, although I think that as a full caster it is often better to stick with your main class as much as you can, to get access to higher level spells.