Yours is a natural reaction, but one that derives from conflating Jorgensen’s surgery with the transgender experience writ large. Before "transgender" was coined and before gender reassignment surgery captured the public imagination, gender-variant people existed whom we would unquestionably label as transgender today.
Money wasn't, after all, working in a vacuum. He gathered his (ultimately disproved) ideas from the experiences of gender-variant people and from his academic peers who were also thinking about gender-variant people, who were actually quite common in queer circles as in Greenwich Village, San Francisco's Castro, Amsterdam, Berlin, etc.
While gender-affirming surgery has become important to many trans people, it isn't an essential part of what it means to be trans, especially in parts of the world like India where gender variance is common and surgery rare.
But even in the US, many trans people never have surgery, some because they can't access it, but many because they don't want it. For them, Jorgensen is an interesting historical figure without a lot of relevance to their lives.
They remind me of Marsha Johnson, who I used to see around Greenwich Village, though I did not know her. She didn't call herself transgender, because the word was too new. She'd never had surgery and reportedly didn't want it.
But she lived as a woman and used female pronouns. Hers is a transgender reality not well understood through a lens of Money/Jorgensen, but well explained by Butler and other theorists.