He do admit that there is currently no satisfactory solution to the Born Rule issue, yet he has written several posts talking about MWI as it is "obviously true". That is quite irrational.

That doesn't remotely follow - at least not without a rather antagonistic interpretation of Eliezer's position. Eliezer is clearly not claiming that there is a theory that gives a good explanation for what causes the Born Rule to behave as it does. He is just claiming that supporting a theory that tries to pretend there is just one world given what we do know about physics would be batshit crazy.

and the fact that we got no quantum gravity solution nor any ToE should force even the most stubborn MWI'ers to keep their minds open and refrain from claiming that it is true.

To precisely the same extent that the a lack of a quantum gravity should oblige people not to affiliate with general relativity.