It's up to the DM
Both interpretations are valid. On the one hand, when you carry someone and you slow down your own speed, the other person's speed will also be slowed. On the other hand, despite of the name, the class ability's description says that it allows you only to reduce falling damage, not to slow your speed.
The nature of this damage mitigation process remains vague, thus, a subject to DM's interpretation (see also What is the source of the "spells do only what they say they do" rules interpretation principle?). DM have to make adjudications aside from the rules, that's what we have DMs for. The Slow Fall ability works differently with different DMs, it is completely normal for the 5e paradigm.
As a DM, you should allow this
The main DM's job is not following the rules as strict as possible. The main DM's job is ensuring that all the players at the table had a good time:
As the Dungeon Master, the most important aspect of your role is facilitating the enjoyment of the game for the players.
Always follow this golden rule when you DM for a group: Make decisions and adjudications that enhance the fun of the adventure when possible.
-Adventurer's League Dungeon Master's Guide: Additional Tips for the Dungeon Master
(available from the Adventurer's League resources page)
The rules is a tool for you, not a straitjacket. Your priorities should be: Fun > Story > Rules.
Grappling is underused in 5e. Your player has an interesting idea for their unique fighting style. He/she puts some efforts into the character, you shouldn't throw the idea out afterwards, that would be unfair.
...unless it breaks the game
However, you are supposed to care about all the players, not just about a single one. If one player starts doing too much with their character, other players can't contribute on par with them. That's why we say such a character becomes "unbalanced".
For instance, if a character (due to rules misinterpretation) gains a possibility of doing too much damage, other damage dealers would feel useless. Feeling useless is not fun.
Nevertheless, I don't think this Zangief-style Aarakocra monk is unbalanced. There are already a few restrictions for this style:
- You can do this only with light targets.
- Your speed (including your flying speed) will be halved while you maintain grapple.
- You either deal little damage, or you can damage yourself in the process.
- If you take damage, you fall prone. You will have troubles with the subsequent moving.
- You can't deal with multiple foes like that, and you probably trigger opportunity attacks.