I've only found a few wild types with extended picking seasons, most ones I've known have very short seasons, which is a trait favored by those selecting for commercial crops.
If you find high quality ones, you can always multi-graft a tree to allow for extended season, something I've seen a lot of with loquat growers.
For recipes, I think anything which highlights the excellent texture and mild flavor of a loquat is a requirement for showing how well balanced a fruit it is, so it should have no extremes or added flavors ideally. My favorite is loquat in agar jelly, the jelly isn't too firm and it holds the loquat flavor well, and only seems to preserve the texture. Not sure this is something you could buy stateside, but if you travel to Japan, it's available in convenience stores when the fruit is in season.