Melee Enemies might become pointless
You're correct that Spider Climb is, in almost every way, a strictly inferior version of Fly. Fly gives more freedom of 3 dimensional movement, gives (most) characters an increase in speed, and can be cast on multiple creatures with higher spell slot usage. The only ways that Spider Climb has an advantage over Fly are a longer duration and a lower spell slot cost. This later feature is important for two reasons: first, because Spider Climb can be cast by lower level characters. Second, because lower level spell slots can be used more often. It's this second feature that creates a potentially unbalancing issue with your idea.
DnD 5e is balanced around a party of four characters (see DMG, p. 274 Expected Challenge Rating for evidence of this). Fly would require the use of a 6th level spell slot or higher to get all four characters in a party out of melee range. But Spider Climb (assuming it scales similarly) would only require the use of a 5th level spell slot to do the same thing. This distinction may seem minor, but there's an important distinction between a 5th and 6th+ level spell slot.
It's much easier to replenish 5th level spell slots
Many classes have features that permit them to regain spell slots. For example, Wizards can use their Arcane Recovery, Sorcerers can use Sorcery Points to create spell slots, and Warlocks can simply take a short rest. These methods mean that characters can cast certain spells several more times every day than their initial spell slot allotment would allow. However, all three of the features mentioned above will only replenish spell slots of 5th level or lower.
Spell slots of 6th level or higher are more precious. Until you reach extremely high levels (19th +), they are each only available once a day. A strategy that requires such a spell slot would be difficult to do repeatedly, and quite expensive. So if you wanted to get your entire party out of melee range every combat it would be difficult at most levels to do so with the spell Fly.
However, if the spell Spider Climb was adjusted as you've suggested, the entire party could potentially be taken out of melee range every encounter. A warlock could get a 4 person party out of melee range for two continuous hours if they could maintain concentration. And they could do so multiple times a day taking short rests.
Of course, a warlock would also be able to get most of the party out of melee range using the Fly spell as written. But there's a difference between getting the entire party out of melee range and getting most of them out. If a single party member is still on the ground, then all of the melee enemies can focus on them at the same time, making the idea of the remaining three staying at range and chipping away at the enemies' health problematic.
Still... it's a viable change
As written, Spider Climb seems to exist to be replaced. It's a 2nd level spell that is strictly inferior to Fly in almost every way. Many classes (like Warlocks and Sorcerers) have the ability to replace spells known as they level up, and Spider Climb is a great candidate for such a replacement.
But your proposed change would make Spider Climb relevant at later levels. And there's nothing wrong with that! There are lots of scenarios where the spell Spider Climb would remain balanced with your proposed changes. For example, if you capped the maximum number of affected creatures at 3. Or if your party is has more or fewer than 4 characters in it. In these scenarios, the change would make Spider Climb a much more welcome addition to many spellcasters' collections (and might be an especially welcome change to casters who can learn it but can't learn Fly, like certain Circle of the Land Druids).
There are also a lot of scenarios where the spell Spider Climb wouldn't cause too much trouble. For example, if the ceilings in a room were 10 feet high or shorter, most enemies on the ground could still reach enemies who were walking on the ceiling or walls. Similarly, if there are no vertical surfaces available at all, Spider Climb is will not permit characters to easily stay out of melee range. If these scenarios come up often in your game, you wouldn't need to worry about the Spider Climb spell being active at all times, and could still design encounters with melee-centric enemies.
I'd advise against your proposed change in a party of 4 in environments with many high vertical surfaces. In such a scenario, your proposed change could introduce some balance issues. But outside of this very specific scenario, your proposed change is entirely viable.