It's not unheard of[1]. Ancient Athens used random selection for many offices and some Italian city-states like Venice and Florence incorporated elements of randomness. A lot of historians think that it did help a lot in reigning in corruption and making it harder for foreign powers to influence their politics.

I think randomly selecting a president is probably a bad idea but randomly selecting a parliament and then having them elect a prime minister from within that group would work well.

[1] https://en.wikipedia.org/wiki/Sortition