Of course, that is half the point. If a player wants it, they get a handout of Common Cold Spells. And maybe a couple common spells from their race, background and homeland. But they will never get a handout of "all the cold spells".Now, the OP mentions having a bunch of themed spells (like cold spells), and I would hope that if a PC has a cold-theme such as a Silver Dragonborn Sorcerer or something, the DM would allow that player to view these cold-themed spells for that PC.
The vast majority of my games are the exotic adventure type....as that is the sort of game I like. Having the PCs head off into the unknown is much more fun! Seven miles underground the drow foe has shadow bolt and acid strike...not the PH spells.That's cool, but that's you. This DM has many of these spells (presumably) to give the players "new magic" to encounter during gameplay. It adds to the mystery and wonder and I understand that. Especially spells which come from species or regions foreign to the PCs.
The players I mention are players: they complain but still play in my game. The spells is not enough to make them walk out of the game.. And if a DM were to have access to a thousand pages of homebrew spells, I doubt that they would do as you just mentioned. That's a good way of losing your players to another table and harming your own rep as a DM. You want the players to trust your skills as a DM.
Not that I don't have plenty of houserules that do that....
Convert all the 2E spells into 5E? Hummm, who would have thought of that.....Hey there's a reason I keep all my old books around. Sure, a lot of what was written in the 2e era was "throw stuff at the wall and see what sells", but for all it's flaws, there were a lot of really great ideas that came out of that halcyon era.