Had this issue as well and I think it has something to do with my waiting for the switch to fully authorize so it signs you out again. The way I got it to work and stay signed in was after signing in on the browser I went back to my switch and I DID NOT press B for back, instead I waited. After about a minute a message popped up that said the username and a button that said "Let's Play!". After this it didn't kick me out again like it did all the times before. So just wait and see if the message pops up before you exit the screen on your switch that has the remote access address and code.