A combination of them being good at skills and their reputation (stereotypically at least) for not taking things too seriously. If bards had no magic, I feel like my bard could still get by and if there was something big he wanted magic-wise, like a Wish, he could potentially eventually get that through his relationships.
Everyone is good at skills. That's what bounded accuracy does. Bards are better at skills, and that's limited to a slight increase for many with a decent increase for a couple. Anyone can build relationships. You should just make a rogue and pick up the musician feat at 1st level. Does what you want better without impacting other players' bards.
It's also never been a thing in DnD, and doesn't match up with the myths and legends on which the class is based.
As it is, there's little reason to play a bard for skills because rogues are better at it, and only lack the small bonus from jack-of-all-trades that's been delayed to a 5th-level ability. Rangers start with the same number of skills and expertise earlier. Expertise is also a single level dip away for other classes who can add a skill at the same time for the multi-classers.
DIET isn't bad for some spell access (it gets tight at higher levels though), but it's the worst application of spell access for bards in any editions so far. A variety of spells from different caster classes part of the class identity. Adding abjuration and selecting spells from the arcane, divine, or primal lists would be a very easy fix for that spell list access. I prefer the choice of which casting list and having divination and enchantment plus 2 or 3 selected schools; those can be 1 addition school of choice at 1st, 5th, and 11th level. That feels like customization withing spell preparation and probably still underpowered. The point is that lack of a versatile spell list negatively impacts the the class draw significantly.
But why would a player play a bard compared to a cleric? Similar role, better healing, full spell access, better armor, plenty of buffs available. The skills and inspiration aren't much of a trade-off after slowing inspiration access down and gutting the spell list.
And why would a player play a bard compared to a wizard or sorcerer? At least spell preparation is a benefit over sorcerers, but they have access to every spell a bard does and more in their selections plus metamagic. Wizards can cast every spell a bard can and more often, modify it to create a better version, swap on the fly with the memorize ritual, cast rituals from the spell book, etc. Both sorcerers and wizards have better defensive options via magic and better at will options in cantrips.
A trade-off of a minor skill bonus to most skills and a decent skill bonus to some skill? Skill bonuses aren't that significant due to the number of skills to which they apply.
Bardic inspiration is typically 2-3 at 1st level in the current system and usually 3-4 per short rest at 5th level (9-12/day standard assumption). Now it's 2 at 1st level, 3 at 5th level (that's a massive loss in the ability), 3 per short rest at 7th level (we could get 15/day standard assumption at 8th level instead of the proposed 9), 4 per short rest (12/day) at 9th level, 5 per short rest (15/day; finally caught up to current 8th level) at 13th level, and 6 per short rest (18/day). Plus superior bardic inspiration at 18th level, which is decent.
The problem one of the most iconic bard abilities (inspiration) takes so long to see much use. Another minor issue is that it looks like bards get more inspiration in the end but that's not true. The epic rules for max scores of 30 instead of 20 means bards would have been getting up to 10 per short rest (30+/day) and could spam inspiration. Minor because 18+ over 6-8 encounters and 2 guaranteed per encounter is still lots.
However, that higher ability score limit further marginalizes skill bonuses because ability scores apply to more checks and the actual bonus would be higher.
Skill bonuses aren't much to trade in to and bardic inspiration doesn't really come on board early enough for the class. Those are the things you claim are worth it regardless of spells. That leaves spell access as a trade off.
Magical secrets is not a bad ability. But it's doesn't happen until 11th level and 15th level. One of those is likely to be used on the arcane spell list for eventual access to wish which would give still give bards less access to the arcane list than wizards who would still be casting all the same spells but better and more often. We're getting to level 15 where bards can prep 2 spells from another list that wizards can't access. At 15th level wizards have created modified versions of 1st and 2nd-level spells that can spam at will using spell mastery.
Bards getting access to a few more spells with magical secrets is late and allows using the same resource on new options. Other classes are increasing resources. Some of them by a lot.
The other spell access ability is Songs of Restoration. This is less spells added by other similar mechanics like oaths, domains, and patron spells. It's also spells to which bards already had access. The UA took away something that was available, gave it back at higher levels than before, cost a shared resource again, and in the process removed the bonus healing from Song of Rest. That's another net loss for the class.
So we have this UA bard for 2024 that delayed the variety of spell selection unavailable to other arcane casters significantly, delayed bardic inspiration progression significantly, lost bonus abilities, and is classified under the expertise identity when expertise isn't that strong to begin with. The UA even gave bonus abilities for lore checks to clerics and wizards even though that's another bardic knowledge is another bardic ability lost where bards need to use skill bonus abilities to replicate their own ability.
What you're saying is the strongest ability of a gutted class would be fine if we took that away too? Are you trolling? Do you think bards are just NPC's the players are pretending to play? ;-)
The classes was nerfed too hard in response to opening up spell preparation while other classes were buffed. A class that's a jack-of-all-trades still needs to be reasonably good at enough areas to be worthwhile. Being good at skills but not the best doesn't do that. A renaissance class needs to be capable of developing well enough in multiple areas to be worthwhile even if that's not all areas. Being good at skills but not the best (rogue) does not do that.
I'm glad you enjoy a skill based gameplay style. I think there is room in the game for that without the combat benefits or spells tied to it. I doubt it would be popular but I think there's design space. If that's the bard then it's not just removing spells. There would need to be a much bigger overhaul of the class.