So, I can take the Criminal background but say my character was raised in a temple and grew up totally sheltered from any kind of criminal activity? I mean, ok I guess, but that feels pretty weird…
There are a lot of religions (though as I say that, it's mainly Christianity) which are literally illegal in certain places in the world, and open worship can get your head cut off.
I have very little interest in seeing the "optimization game" for backgrounds, so my current inclination is to go all-in with custom backgrounds.
This is a complete oxymoron, and the main problem I see when people talk about this. They say "I don't want to optimize, I want STOOOOOORRRRRYYYYYYYYYYYYY!!!!!!!1!!!!!!
Then they say they can't do Story because the Backgrounds don't give them the ASI or Feat they want. That's called Optimizing.
Farmers are tough. They work with their hands. They plow fields. They are street smart (Wisdom), not book smart (Int), and they are strong and usually have a good constitution. Nobles are healthy (strength) cause they've been well fed, smart cause they have the funds for education, and charismatic so they can deal with other nobles. These things make sense. If you want Story, these Backgrounds are great with Story.
So you're an Orc who lived in the Temple to Gruumsh instead of going out raiding every day (Acolyte)? You're not going to be as strong, at first, as the one who was a Soldier on the front lines from the moment he could crawl. That's OK.
You're a Rogue who spent time in the academy and became a Sage? Well, you're not going to be as dextrous as the one who spent his time running the streets breaking in to buildings, surviving on his speed and charisma.
This is what Story is. This is what Role Playing is. And in the end, you're talking about 2 ASI points. A 5% greater chance at success than someone else. It really, REALLY isn't a big deal.
Use the ASI to shore up something that class usually has a Weakness in. Or an ability they aren't usually associated with, and role play that. I played a Warlock with a 4 Con and a 6 Charisma. He turned out awesome.
Creativity thrives under restraint. Your characters will be better for this. If you're complaining about not being able to put ALL of your points into the stats that would do the most damage, then you're more worried about optimizing than you are about story or role playing. Because yes, all Acolytes will be weaker than all Soldiers. Deal. Or don't be an Acolyte. Be a Devout Soldier who kneels and prays before every battle.
I am the type who optimizes around a concept. I find a concept, and build to that concept, tweaking until it is as optimized as possible, but I will TOTALLY take a suboptimal option if that option is a better fit for the concept. So if I'm making a Noble Aasimar who becomes a Warlock, I'm going to take Noble even though I don't really need Strength or Intelligence. If I want to make a Trickery Cleric who has the Charlatan background, I'll deal with my Wisdom being a little lower to start, and put the points in Charisma instead so I can deceive better. His Wisdom will get better! And it won't take long! Every single Feat has at least one Ability point. That's HUGE!
Optimizers: You will be ok. It will be fine. Custom Backgrounds sound like you'll have so many options, but you won't. Every single Monk will put their scores in Wis and Dex. Every single wizard will go Int and Con. Period. Every time. Because why wouldn't you? You will never play the Wizard who is more street smart than book smart. Or the Paladin who loves History. Or the Monk who is good with the ladies. Like, there is more to these characters than just their primary ability scores.
