If you are accompanying yourself precise pitch is not necessary. But There must be something that changes the emotional appeal with actual scale. Particularly if there is someone in the audience who has perfect pitch, it will annoy them unless they are a concert muso.

How many people do you hear where the music is strange until you recognise the song, or the second verse starts? And how hard is it to start a song when the person before just brought the house down? Short term memory is strong until there is a loud bang (= danger signal and thus it wipes that memory to deal with it).

Orchestra soloists like Galway & Kennedy invariably play a miniscule sharp - it makes them sound bright.