You're right that most LGB people support trans people. This is also because we are more likely to know them and thus are actually far more educated on the issues than most straight/cisgender people.
I can't speak for cisgender people because I am not one. However, if you think the reasons to limit what access a trans identity gives a person to the spaces and resources of the opposite sex are simply because people don't know enough about trans people or don't have lovely trans friends you are missing the point.
I believe the experience of having a female body in a sexist society is meaningful enough to justify rights, existence and a voice separate to male people of any gender identity.
I do not want a person who has not lived as female from birth taking public or professional roles where they speak for female people. Firstly because I don't, based on the things said by the most visible examples of people who have put themselves in exactly those roles, think those people truly understand what living from birth as female is like, and secondly because female people are already underrated and like any underrepresented group they need and deserve to see people like them take active and visible roles.
I do not want a person who has benefitted from male privilege for 40 years taking prizes and opportunities intended to build up female people because that person has identified as a woman for 5 years.
And none of that is anything to do with fear, dislike or disbelief in trans people. It's do to with supporting female people in our centuries old fight against sexism. And as a female person I sure as hell know what I'm talking about.