what's confusing, to someone trying to logically parse the language, is why anyone would want to write:
if anyone wants to come, they are welcome.

Because it is correct English, and often simpler than the constructions used to avoid the issue (particularly in long descriptive text on user interfaces where the repetitive use of the original noun is a little wearing). It sounds perfectly normal to my ears. But that is why I asked the question in the first place, because I was aware some people disliked the usage, and thought this was a good place to ask. Thanks for the feedback. I may modify my behaviour slightly

Rod