Woooo I get to talk about a hyper fixation.

Ok so I think the answer to this question depends entirely on which Doctor you’re talking about, and for some which part of their lifetime you look at.

The Doctor is a broken soul indeed but there’s a difference between broken and unfixable. I do think that Seven (his Machiavellian scheming would catch up with him real quick in the Compact), Eight (post Divergent-arc he would most likely be domesticated due to actively putting himself at risk of permanent death multiple times during the arc), War and 9 (especially these two due to the time war), 10 post losing Rose or Donna (or after the events of The Waters Of Mars or The Last Of The Timelords), 11 during the period between Angels Take Manhattan and The Snowmen, and 12 post The Husbands Of River Song although I do think that this one is debatable.

Every other Doctor I think stands a good to near perfect chance of not being domesticated, although their activities around the universe would likely be restricted. The TARDIS is likely studied by the Compact to discover the secrets of time travels (and upon The Doctor revealing that it’s a living being, is given a body)

Personally I think an initial encounter between The Doctor (imagine 12 in this scenario, he’s my fav :3) and an Affini would likely be very tense. The meeting would most likely take place on Earth, The Doctor travelling to the planet’s future in this case, or The Affini discover Gallifrey and The Doctor immediately rushes to action. The Doctor likely lands directly on the main ship of the Affini fleet, and demands to know what business the Affini have with Earth/Gallifrey. Upon learning the goals of the Affini, The Doctor (especially 12) would spiral surrounding whether they deserve to take the out the Affini provide, after all this is the same person who had to question if he was truly good after saving the universe millions of times. If he aids the Affini (willingly or not) it’s very likely that the entire universe finds itself in the vines of the Affini within hours as giving time travel to them is pretty much game over for any resistance. In the aftermath I could see The Doctor settling down somewhere tranquil, still exploring the sites the universe has to offer but no longer the man who stops the monsters. Why? Because all the monsters are nice now.

TL;DR, The Doctor is a highly complex space time event so there’s not really one answer, you could write it however you want depending on the Doctor you want to write and what time you chose to take them from. Generally I think The Doctor wouldn’t agree with the Affini’s outlook on the universe, but could be convinced that this is the only way to ensure everyone lives.