PSL does have a Trot genealogy, though is ML. Socialist Workers Party -> Workers World Party -> PSL. SWP was Trot. WWP were an OG tankie split and the origin of Marcyism, literally led by Marcy. They adopted more ML stances but didn't shed all Trotskyist thought. So many people handing out little papers and making unrealistic proclamations about what organized labor would do in the imperisl core. Then PSL split off of that over internal organizational disputes. PSL seems to have been on the right side of that and is obviously now even cooler and more ML, and explicitly so.

But I do think see echoes of its Trotskyist parentage from time to time. For example, putting so much energy into electoral vanguardism, aping Bolshevik structures, and coming across as detached from the community when it comes to local work. To me, one of the more salient failures of Trots is in how they tend to aesthetically go through the motions of an idealized canon of What Trotsky Said To Do, which is often just what Lenin et al developed for their conditions, and generally fail to do the science of revolution. Much time spent self-congratulating, picking bad fights with others while repeating boilerplate lines, and then launching into the Next Big Thing that is Very Communist because leadership made A Great Speech. This does not describe PSL, but I do think you can see legacy bits of that kind of thinking pop up on occasion.