"It might be nice to join them sometime but the person who organizes the trips is kind of old-school and would likely disapprove of my gear."
There's a highly organized outdoor group in the Seattle area, the Mountaineers, that I sometimes hike or backpack with, and I've led a couple of backpacking trips for them. As an aside, leading trips allows you to set the ground rules, at least insofar as those fit within the overall rules of the organization.
Like I suspect a lot of such organizations, the Mountaineers have some old-school, conservative policies built into sometimes documented rules, and sometimes into the heads of trip leaders. Apart from climbing a *real* mountain, I like to use trail runners rather than boots, and this is the most common sticking point. I've not encountered serious issues with much else in my gear, and I did one backpacking trip with them where I used a poncho as my only shelter.
On a moderate to short trip, I might toss a pair of light hiking boots hidden in my trunk just in case they get really sticky about the "no jeans, no tennis shoes" sort of mantra. But I'm always ready at need to "play the thru-hiker card", which you certainly can too. I.e., "I went through the Sierras in June in this type of shoe when the passes are butt-deep in snow, went through the Smokies on the AT in March in nothing but snow …". If done correctly (not pushy, arrogant, etc, just sort of background fact sharing) this can give some "cover" to the trip leader, allow them to say "okay" if they're at all inclined to. I think I've ended up wearing the light hiking boots only once on such trips.
Now, slow pace, taking lots of breaks — that definitely has potential to make a person nuts. I think it boils down to expectations, and just sort of changing process to not worry about that stuff, take advantage of the benefits of group hiking instead, i.e., walk slowly behind someone that you enjoy talking to. When you're up and ready to hit the trail in the morning and others are just beginning to fiddle with gear and cooking and so forth, wander about and chat with folks, have a second breakfast (always a favorite activity for me when food supplies permit — or I'm offered excess food that someone else carried! :-)).

