Cutter to sloop
It greatly depends on the specifics of the design in question. Many cutters are designed where the jibstay is critical to the strength of the rig and so the jibstay cannot be removed. On the other hand many cutters do very will with the jibstay stowed.
You are absolutely correct that it is a royal pain to try to tack a genoa around the jibstay. It is one of the reasons that I no longer consider a cutter suitable for single-handing or even short handing.
The current thinking for offshore work has swung toward fractionally rigged sloops with minimally overlapping or non-overlapping headsails so that you do not need to mess with a genoa and you already have your stay sail up and rigged and do not have to go forward and rig a forestay and hank on a jib when things get dicey.
Good luck in your boat search,