I often see people touring on flat bar hybrid bikes, the sort of bikes that you might otherwise expect to see people commuting on. They rarely come from the manufacturer with a fork that can take a front rack but otherwise the people that I saw riding them were having no problems at all with a rear rack and some form of handlebar bag or basket to get some weight on the front. If you could pack light and get all your stuff on a rear rack, they appeared to work ok. I would be hesitant to ride with a heavy load all on the rear rack, but if you can keep it light it has been quite doable based on what I saw others doing.
Some of those hybrids had gearing almost as wide as many touring bikes.
The photo is of one of the riders I met last summer, she had toured from Toronto to Cape Breton Island where I met her. Note that she put some of the weight on the front of the bike too.
And another hybrid rider, she was from Germany where a lot of the touring bikes have flat bars. I rode with her for a day and a half as our routes were briefly on the same roads. Her bike was the one on the right in the bike rack on the ferry. She had more weight on the rear rack than I would have wanted when there was only minimal weight on the front, but the bike worked well for her.
My point is that there are a lot of perfectly usable non-touring bikes that you can tour on that meet the criteria you specified.
I met several others on hybrids this past summer that I did not take photos of the bikes, but they were doing quite well.