Is this what they meant by “be gay, do crime”?

Native Spanish Speaker here - In a nutshell, yeah that’s pretty much it! =)

To the best if my knowledge, Santo is used to clarify the difference between the title and the name.

Santo Tomás being the simplest example I can think of, as “San Tomás” can be confused as as “Santo Más”.

Everything else is pretty spot on and an excellent explanation!