Limits Make Special Sense for Adventurers League
Adventurers League has several unique challenges:
- Adventurers League DMs may have different players with different
characters dropping in for each session, so they don't want to
encourage too many of the abilities of those characters to come from
outside core materials which everyone is presumably more familiar
with. It minimizes the number of unfamiliar things that have to be
ruled on (and without making up house rules, because this is AL).
- They also have to have adventures that work and are balanced for
whatever characters they allow to show up. The Aarakocra race is
off-limits because it can fly, which can trivialize certain
challenges they might want to include in the game, and
disproportionately reward players who solved them due to this
- They also have several campaign settings and limit certain materials
to the campaign setting it was designed for primarily for reasons of
lore (no Warforged in Faerun, etc.).
- Finally, AL games are open to all experience levels, and new players
are often intimidated enough by just the PHB options.
In this context PHB+1 makes a fair amount of sense, but it is what we might categorize as a "big dumb rule" implemented to minimize having a long list of specific rules on character design for players to navigate and DMs to remember.
Limits May Make Sense for You, But Not Necessarily Those of Adventurer's League
As alluded to above, it is unlikely that you have a drop in drop out system for players (though I play in a non-AL campaign that is run that way subsidized by a game store). It is also unlikely that you are bound to do an adventure in a set way that does not adjust to the characters people have actually brought to your table. There are a few people who would do it this way, but they are a distinct minority.
Player characters being balanced with their party seems to be a concern of Adventurers League. Many home tables could not care less about this, or at least will happily tolerate a modest imbalance. You also may or may not have a wide variation in experience among players.
So yes, it is possible that the AL reasons for limiting options apply equally to your home game, but many of them probably do not for most non-AL games. Putting your game in a box designed for Adventurer's League may or may not be a good fit.
Placing Limits is Potentially Far More Trouble than it is Worth
There are several excellent answers pointing out the advantages of placing limits on options and I don't think I can really improve on them, but for a well rounded discussion we should also address the drawbacks and advise caution.
Don't break your players' hearts if you don't need to. Especially if you are just meeting with the person for the first time "you can't do X because I don't want to deal with learning about it" may make for a bad first impression. Even worse is if they get several levels in not knowing you will not allow the subclass or feat they have been planning for, or a signature spell they are building around. Also they may have just dropped $50 on a book that gives them these options and not be able to use it.
Perhaps more importantly, you are potentially creating more work for yourself in terms of knowing all content available if you try to actually tailor what is allowed to your particular preferences. Even going for PHB+1 doesn't really narrow what you potentially need to know much as long as that +1 is anything and up to the player.
The simplest solution (other than "big dumb rule" limiting options) and least contentious (other than just allowing everything) is to just ask players who do want to bring something from supplementary materials to let you look over it and make sure it doesn't have something that will obviously break your campaign, seems overpowered, etc. For example, I might not allow a changeling in a campaign that was all urban intrigue; at level one they get an at-will shapeshifting ability that is a mostly better version of a late game at-will alter self casting option for druids and warlocks, and in such a campaign being a different person instantly with little possibility of detection might be huge. But the changeling I play in another campaign set in wilderness is at level four and has had no occasion to use the ability except for flavor.
Many players bring in characters built using D&D Beyond and don't actually know what resource the options came from (even if they meant to only use PHB options). This is especially common with new players. Trying to suss out where they got what can require a lot of googling or an encyclopedic knowledge, and can put a player on the spot to pick something else they haven't really thought about.
Also, remember that letting a player steamroll an occasional encounter because of the awesome ability they have is not the end of the world. It can make them feel awesome, and characters should be allowed to just be awesome sometimes. Keep in mind that they have basically always passed up some other options (and hence made sacrifices) to have that ability, and thus been weaker in other situations. The cool multiclass they have might have involved a slog of being underpowered for 4 levels or whatever, in which case they bought what they've got. Everything has an opportunity cost. Aarakocra flight is probably the most cited "overpowered" or "gamebreaking" option, but the race doesn't get a whole lot else and when you knock them out of the sky that falling damage is going to hurt.
There is no option available to my knowledge that will be useful in every situation, and few at most that a DM can't nullify for many or most situations.
Probably the least contentious situation for limiting options is races (or less often classes or spells) that are incompatible with the setting. Not a lot of players expect to play a Tabaxi cat person Monk in Middle-earth, and limits may help a lot to get away from the "kitchen sink fantasy" approach of default D&D, which many people hate. Note further that even published materials that are for particular settings (e.g. Ravnica) are often not intended by Wizards of the Coast to necessarily be used in other settings. Limiting things to the more general supplements (the main ones for player options being Volo's for races and Xanathar's for subclasses, feats, and spells), seems to usually be less disappointing to players.
Finally, if you have a player with the knowledge to exploit supplementary materials for some sort of overpowered monstrosity, they are probably even better versed in how to make comparable powergamey monstrosities from the more familiar PHB materials. Beyond this, many if not most supposedly "broken" combos come from feats and multiclassing which, despite many people's expectations, are themselves optional rules you don't have to allow.
Personally I find the prospect of trying to enforce limits beyond "run it by me" more daunting than a free for all, but other people's mileage will vary.