First things first
This Training sub-section is under the Other Rewards section. Read it carefully:
As much as adventurers desire treasure, they often
appreciate other forms of reward. This section
presents a variety of ways that gods, monarchs, and
other beings of power might recognize the characters'
accomplishments, including supernatural gifts that
give characters new capabilities; titles, lands, and other
marks of prestige; and boons that are available only to
adventurers who have reached 20th level.
I.e., this is obviously assumed to be a reward given by the DM, not a player training each other. Using it that way is not intended.
Role-playing wise, the easiest explanation is: PCs aren't that skilled. This might be a hole if you are talking about 11+ level characters, but check the description of the trainers:
The trainer might be a reclusive wizard or
haughty sorcerer who owes the queen a favor, the
knight-commander of the King's Guard, the leader of
a powerful druid circle, a quirky monk who lives in a
remote mountaintop pagoda, a barbarian chieftain, a
warlock living among nomads as a fortune-teller, or an
absentminded bard whose plays and poetry are known
throughout the land.
None of these fits a low level character. The bard in a low level party is barely known in the local inn. The Barbarian is now starting to get used to brawls. The Wizard has 10 spells on his one book, not a tower with a library for himself. They are not the high skilled trainer that would be able to train others.
As for the balance
While I see KRyan's point, I don't think training is that problematic, balance-wise, if well handled. Feats are on par with +2 ASI, which is what you get from items like Tomes and Manuals - you may reward a feat when you would reward one of these tomes, instead, for example.
As mentioned by András in the comments, yes, some feats are stronger than simple ASI, but IMO, that's a problem with the feats balance to start with. The players will be getting these feats, you reward them or not, unless you are not allowing feats in your game. Rewarding the feat instead of the tome simply changes the order when they get the feat and the ASIs that they would get as well.
Also, it's worth noting that this section of the DMG is more of a "inspiration/ideas for you" than actual rules. Note, for example, the lack of how many days it would take to train, how much gold it would cost (if any at all), or anything telling us what would be a good actual reward. What I mean here is: don't take this part too seriously without thinking if your campaign needs it or if it is able to actually support it.