Yes, harder warmup. I've found a couple of full throttle 90 second efforts helpful. I'm another older rider who warms up slowly. It takes me a good hour to warm up. OTOH, I don't get a quad burn. My legs just very gradually don''t work anymore. That takes more than a couple hundred miles though. Good note on using lower gears. If I'm going long distance, I gear down and spin the hills, definitely not popping over them.

However I think you're talking about group rides, so except for cadence, this advice is all useless. And on cadence, on a group ride, best to spin the same cadence as the riders who are in front of you. Sometimes our group rides go right up a hard, competitive hill within a mile of the start. Just have to suck it up.

Physical fixes: besides cadence, use your whole leg. Pedal circles, which doesn't mean pull up in the back. Push forward at the top, try not to push down, pull back at the bottom, mostly unweight the up pedal, but don't pull up. Try to be conscious of not particularly pushing down. Relax your ankles. Relax the muscles which aren't working. On a too-hard ride, your hams should cramp about the same time as your quads. Which sucks but it's better than the alternative. So if your quads hurt, pedal with your hams. When your hams hurt, switch back to your quads.

Chemical fixes: I think what's going on is that it takes some time for your blood vessels to open up and carry away the waste products. You can speed that up a bit with supplements: 1g. sodium nitrate + 3g citrulline malate + 1g beta alanine - every day, whether you ride or not.