I think everybody solves this in their own way.
In my case in particular I don't play at any high-falutin' places, and the churches and other places I play are happy to have someone with any sort of training at all. They're willing to overlook the occasional mistakes, gaffes, and uninspired performances, and they still pay the bill and ask me to come back. I'm not in this to be any sort of great concert pianist, because that requires a set of tradeoffs I am unwilling to make. I play as well as I can and choose venues where they're happy to have me.
There's a balance: you don't want your playing to get sloppy, but you don't want to get so hung up on perfection that you get your head all messed up. I take care of the balance this way: perfection is something I seek when I practice. When I'm playing for people, I'm there to connect with them and with the event, and at that point, there's not much I can do about the quality of my playing, other than to be sure I'm not letting my mind wander. So I listen, and look at the people I'm playing for, and try to just stay in the moment.
Much of the music I play is chosen by others, and sometimes I have to play it on short notice, so if I'm playing something that I don't like and haven't been given enough time to practice, I just do the best I can and if I drop out the alto voice halfway through then so be it.
Everyone has their method.
If you're just starting out the most important thing is to play for a non-judgemental audience until you're accustomed to playing. Sometimes that's hard to find. I think the music education process as it exists now focuses overmuch on competition. You can pursue excellence without constantly comparing yourself to your peers. Playing in front of people, there are constant reminders that there are people listening. Cough, door close, paper rustling, whispers, giggles. You have to get used to that, and get used to staying in touch with the world around you while you play. At least, that's how I do it. Some people build up a sort of a wall of concentration and it's like nothing exists but them and the piano. Maybe that works for them, but not for me. At least, not for leading congregational singing.