1. Setting up a mail server and keeping it running isn't easy 2. Spam filtering 3. (already mentioned) uptime - you don't want to lose email 4. The security of your email doesn't depend only on you - it also depends on where people you are communicating with are hosting their email.

It's much easier to get search privacy since the only other entity involved is the search engine. Which is why I use https://duckduckgo.com/ for search.