There are maybe 5 teams looking for a goalie around the league, and those teams were pretty easy to spot last year. Not all teams are looking to trade for and extend a goalie at a high price. The market for Ullmark was probably 5 teams, and they were easy to spot last year as teams that may be looking to upgrade in net.
Not all teams in the Western conference would be on his list, because you can be pretty sure some teams won't be trading for a goalie because they have their starter already (Ottinger/Hellebuyck/Demko/Saros/etc.) or are not teams looking to compete right now by trading value for a goaltender (Ducks/Sharks/Arizona/Seattle/Blackhawks/Calgary). You could have safely added 6 Western teams and felt confident.
Colorado
Vegas
Edmonton
LA
St. Louis
Minny
That would leave 10 teams in the East you can add in.
Then you look at: Shesterkin/Sorokin/Bobrovsky/Vasilevsky, that's 4 teams you don't need to add for sure. He's already walked from the Sabres, and they have UPL and Levi as young goaltenders they are giving the reigns too. Can add Carolina because they have been upfront about not paying their goaltenders.
Ottawa
Toronto
Montreal (doubtful they'd be looking for a golaie, but you can still add them)
Columbus
Flyers
Devils
Pens
Caps
Canes
Wings
Give or take a team from each conference.
It would have been pretty easy for Ullmark and his camp to create a 16 team list essentailly blocking every percievable trade partner. There aren't many teams looking to trade for bigtime goaltenders