Is there anything preventing some random developer from releasing a native iOS GMail app? I basically just want an "Archive" button and for it to always be in-sync with the cloud. Does anything like this exist?
If you configure as Exchange the emails are pushed to you instead of pulled. Use m.google.com as the Exchange server. Also, the trash icon doesn't actually delete anything. It sends the message to your archive. This is the preferred way because it is push and also syncs the contact list with Google.
Alternatively, if you configure the included client as Gmail it will use an archive icon instead of the trash can, but it is not push enabled. You will have to pull every 15 minutes. You wouldn't get the contact syncing.
You may need to recreate your email account to get the behavior you want if you you made it as IMAP before Exchange and Gmail were options.
Since iOS 4 (I think), if you use Apple's Gmail preset in configuring email on your iPhone, it automatically changes the delete button to an archive button.
You can also manually setup your Gmail as an IMAP account and configure the delete button to send messages to Gmail's archive folder (this is useful if you have added custom email addresses to send from in Gmail)