I think that really depends upon what time of year you are going. Recently, over spring break, it seems that there were a few days where it was so busy that no one without a PS could be seated at the parks. Obviously that is not the norm for Disney though. My family has been going to Disney for years and has never done PS's before. Honestly none of us knew we could do these until the disboards which I just discovered a few months back. That being said, my parents were able to take me to quite a few character breakfast's at Ohana and the Crystal Palace, all without PS.
However, the trend is that more and more people are doing PS. Because of that, the chances of getting seated now are a lot more slim for character meals unless you have a PS. My DH and I are a lot like your DH. We HATE being on timetables. Truthfully timetables don't work for us, they just get ignored. For our honeymoon trip this May we did schedule 4 PS ressies, 1 at a popular restaurant there and 3 for character meals, because we wnated to be sure to be seated at these. So basically, for the more busy restaurants like Le Celier, Cinderella's Royal Table (90 days out you HAVE to have a ressie for this), Coral Reef, etc. and for the character meals, you should have a PS to make sure you are seated. If you can't make it to the ressie or decide not to go don't sweat it, simply cancel it while at WDW. Have a blast!