Dude, LGBT isn’t just “trans people”. nobody is arguing that LGBT ppl aren’t human beings with feelings, but they are easy scapegoats (alongside immigrants etc) to distract people from real problems… unless you consider LGBT people or immigrants or whatever to be problems at all, which are not hence the “distraction”; I wouldn’t be surprised if Putin personally didn’t give a fuck about lgbt people
Nazi germany is a very poor example for your point since Jews, LGBT etc were easy scapegoats for the terrible situations they were in after WW1. You don’t even have to go that far either, look at US, Italy etc that have gone (far-)right and the discussion is always about immigration, LGBT people and other stuff that makes no sense and is either a very small problem or not a problem at all
tldr: governments use easy answer to redirect people’s anger towards made-up problems, so that they don’t really think how fucked they are