Apologies if this comes off as too explain-y and mentions some of what you already know. I am a queer history nerd and I don't get many opportunities to trot this out.

A lot of the reasoning of the shift to transgender came with the transition away from focusing on the strict sexual characteristics of the people involved. Trans sexual was seen as either implying heavily the medical involvement of alteration of sexual characteristics was always the intended goal or defining factor that disqualified people from being properly trans or referenced the highly discredited and now generally considered transphobic connotation of someone being sexually attracted to the characteristics of the opposite sex so much that they they treat transition as a fetish which had "moral" considerations. Basically the whole auto ando/gynophillia stuff. They varied their approach based on whether they thought you were in it for kink and were generally more lenient if you were trying to model what they considered heterosexual norms.

Gender was selected as the more blanket friendly term which applies to how someone self conceptualizes themselves. This does include in it's definition gender euphoria and dysphoria so by it's definition it featureshow one feels about their personal physical sexual characteristics... It just places zero emphasis on how one chooses to respond to those forces leaving the door more open to a wider range of different transition presentations including purely social ones.

It's less a euphemism and more a widening and restructuring to shake up the old harmful preconceptions that existed in old DSMs...it also had a particular historic use for trans people.

Functionally some of the lesser known history is it had a temporary practical purpose of providing red flags for patients of medical and psychiatric professionals who remained out of date to the rather durastic changes to the DSM that retired notions of sexuallity and attraction as a set of Freudianeque assumptions to the underpinning of behaviour that happened between 1990 and 2013. Basically if the doc was still using the term trans sexual you knew they were probably making a lot of their recommendations and limiting your choices based on whom you were sexually attracted to. If you knew your doc was not keeping upto date it gave you some level of personal advocating power in a system regularly stacked against trans patients.

Regardless of how one personally feels about the term it is not a euphemism.