I'll take the official Gmail app over the native Mail app, every day with no questions asked! As stated above, the search function is amazing, compared to the "so-called" search function in the native mail app. It has never ever worked for me - not even once. Searching for a specific mail in the iOS mail app could give me headaches. It'd end up of me logging on to my mail account using an internet browser and from there searching for the mail instead.
With the Gmail app, all those problems are gone with the wind. It's a wonderful app, especially since the search function is working as intended!
I suggest you give it a try at least, and from there make up your own mind.
And by the way, if you can receive the mail you just sent to yourself, everything is working as intended. You'd never be able to receive the mail and then read it if the mail server never received it, and from there sent it to it's destination (you, in this case). So mail away, they'll all be delivered as intended.