Although in theory you could expose your desktop computer to the internet and have the iPhone connect to it, I would view that as a security risk. Better to connect your phone to Gmail.

Do you connect to Gmail using IMAP? In that case, Thunderbird, your phone, and Gmail online would all be in sync. If you use a POP connection, it will be trickier to get a consistent view across platforms and avoid possibly stepping on one another.

Regarding how best to access Gmail on iPhone, you might check for opinions on iOS forums also.