Great pictures and sorry you have not had perfect success, but hey, that's gardening. We try and sometimes don't succeed, but hopefully we learn. At least you are doing well with most of those crops. Peas and chard, though, hmmm...those should be easier than some of the other plants that are going well for you.
My questions:
(1) Were there any birds in the area big enough (crows?) to eat pea seeds? Squirrels/chipmunks? Is it possible the seeds did sprout, but that something immediately ate the foliage that peeped up above ground? Rodents of any sort or raccoons or birds are possible there.
(2) Where you live and during this period, did your garden receive rain or did you water it thoroughly so there was moisture in the top few inches of soil? Seeds can't be allowed to dry out or they won't sprout. That's a fairly loose soil mixture you probably have (aside from the clay content), so not sure how long it holds the moisture.
(3) Did you buy fresh (this year's) seed from a reputable seed company? Chard seed normally lasts a few years, but the germination rate with peas and beans goes way down after one year. If you bought these from a seed rack at a store, it is possible they were sitting there for a while. Some stores will keep them on a rack for years, literally.
(4) Soil composition sounds great. You could get it tested if you want, but even if it were (just for sake of argument) imbalanced in terms of pH or nutrients or contamination of some sort, the seeds should germinate in it just fine. The others did.

Those are some initial thoughts.