Yes, they need to compromise the server but they also need to compromise the client. With both compromised they will have the same execution privileges as the original Putty.exe, then they will need to ROPgadget? If they have both compromised by the heck do they need to use putty?
An attacker might just need to get something to show up into a log file that is then viewed using PuTTY. Always escape attacker-controlled data before logging or displaying it.