More importantly, Crist not only switched parties; he switched IDEOLOGIES. He wasn't a conservative Democrat who switched to the GOP; he was a conservative Republican who moved to the center somewhat (but not totally), then switched to being a neoliberal Democrat. THAT'S the part that some people just can't get by. He was known as "Chain Gang Charlie" as a Republican, then he became a Republican, then an Obama Democrat. The only person I can think of who took this path was Goldwaterite Rep. John Hall Buchanan (R-AL) who ended up being founder of People for the American Way. His party switch was too cute by half, and THAT (sadly) is what kept him from ousting Rick Scott in 2014.
There have always been things Charlie was consistent on, though. He's always been good on the environment, which is something people in his district care about — he was running on fighting offshore drilling when he ran for Senate in 1998. He's usually been good on education issues too (two of his sisters are public school teachers). There are lots of things he's flipped on, but there are some through-lines, and they seem to be the issues he may actually care about.