Actually, ICMP-based PMTUD is almost dead in IPv4 due to this exact problem (since ICMP isn't a "protected" protocol which is required for IPv4 connectivity), most actual services tend to do the MTU discovery purely using UDP or even using TCP (https://datatracker.ietf.org/doc/html/rfc4821)
That is essentially an reaction to random middleboxes just plainly droping ICMP traffic. If you want to stuff to work you do not want to just drop ICMP. The sane policy is to just pass it through or maybe rate limit it.