I installed 10.13.4 today and the problem is much, much better now: keychain now finds items with acceptable speed without showing the spinner.
However, I think they just found a workaround and I am still convinced that this is a hardware problem. When I try to see reveal passwords in the keychain (check this checkbox and then you get prompted for your computer password), it works only in one third of the attempts. Many times, if not most of the time, the password does not show up and I have to re-check the box and re-enter the password. Sometimes it works, sometime I have to enter the password multiple times. It seems to be better when clicking the button instead of entering using the enter key.
My interpretation is that in the background the same timeout due to some fancy touchbar-secure-enclave thing is still there, but they managed to remove the blocking UI with the spinner. For most of the time it looks like they misstyped the password, so most people probably don't bother and just enter the password again.
So far so good. It is useable at least and a lot better than before. I guess moving to the next MBP (without touch-bar) will be the final fix for me.
Did the update fix work for you?