FrogReaver
As long as i get to be the frog
You can already by core RAW effectively take any starting skills you want in almost every instance.
1) Background skills are officially just suggestions. By RAW you need DM permission to play a half-elf, but you don't need DM permission to take any 2 skills you want (and otherwise customize every part of your background other than making up a new background Feature). If you don't have Perception, it's because there were at least 2 other skills not on your class skill list you wanted more.
2) Class skills lists are rather generous. Really, the only ways they should be a problem at all is if you are ignoring RAW on #1, or you are making a character that is taking all skills that are weird for their class. I suppose if you were leveling dipping at 1st for mechanical reasons and your class concept was based more on your second class, that might possibly come up.
You can almost always just pick the starting skills you want from the whole list, without looking at the class list, and then go back and look at the class list and see that your build is valid. I have never seen a character not get exactly the skills they want RAW. So sure, maybe that's a reason to remove the rule--since it basically almost never comes up. I expect it's there to help new players.
So I see about 2 situations where there is an actual limitation. If you aren't playing by RAW and can't pick whatever skills you want for your background (way too many people miss that rule for some reason--it seems like more people miss it than know it!), or if you are mechanically dipping first level in a class themed completely incompatibly with you character concept.
So here's a challenge for anyone: Think of 5 or 10 character concepts you or a friend might want to play who wasn't trying to make this not work, write down which starting skills you want them to have, and then crack open the book and see how many of those characters aren't completely legal already. I'm guessing that you'll get all legal options unless you are intentionally trying to make a non-legal character (and depending on your memory of the books, you might still make a legal character when you're trying not to!)
If the restriction is avoidable in practice then that's just further evidence it can be removed completely. If nothing else it allows you to pick a background based on your character rather than the skill benefits it provides you. That's a win in my book.
I think your challenge is a bit biased because we are so used to certain standard d&d characterisms that we are prone to pick skill combinations that are going to work anyways.
My first: a fighter that's a diplomatic spy. Important characteristics, talking to and reading other people to get what he wants. Being able to pick locks to see what people are hiding when he finds an opportune moment. He's a trained fighter as a last resort.
To me the most important skills for him would be:
Persuasion
Deception
Insight
Thieves Tools
And I know the accusation will be that this is a character that I designed to not work - but the thing is I can design 100's of such characters - and they are all characters I never think about because I know they won't work in the 5e system. So whatever inspiration I may have had is getting instantly filtered out.