One mild-flavored perennial that doesn't require a bunch of special preparation is Erba Stella (aka minutina, buckhorn plantain, etc. Latin name is Plantago coronopus.) I seldom see it mentioned but it is mild enough to be included in high-end salad mixes, and has a crunchy, texture and slightly salty flavor that makes it easy to use and like. You just cut it and throw it into salad, soup, etc. It is hardy to down to zone 5.
https://www.quailseeds.com/store/p261/Erba_Stella%2C_Minutina.html
Perennial/biennial/annual aren't hard and fast categories. Several of our common vegetables are perennials in their ancient homeland, or in certain climates. Tomatoes and peppers are often perennial in fairly frost-free climates. Runner beans and jicama both make tubers that overwinter in mild climates. As you get into colder winters, the number of possibilities decreases, but one promising place to look is at plants that are normally biennial. These are plants that overwinter, then flower and die, usually around midsummer. Cabbage kale, onions, beets and carrots are all biennial vegetables. They already have the hardiness to overwinter in the normal course of being biennial. So from overwintering once to doing it several times is not such a big step. Often they have ancestors that were perennial, like Sea Beet, the ancestor of beets and chards.
If perennialism is an important trait to you, it's well worth experimenting, especially with crops that are traditional in small homestead or cottage situations. In situations where the field or garden is all plowed every spring and replanted, then weeded or cultivated, there is selection pressure for fast emergence of seeds, fast maturity, and annual or biennial habit. This is the kind of evolutionary pressure that turns a perennial like Sea Beet into a biennial like table beets. In more relaxed garden situations, where trees, shrubs, flowers and crops are jumbled, the traits for perennial survival may survive. If you want to find crops that produce palatable food in usable quantities, a good place to start looking is among domesticated garden plants that have not been as intensively cropped.
Kale is an example. The varieties that have been developed for farms, (like Vates, developed by the Virginia agricultural experiment station,) rarely perennialize, even in maritime climates. On the other hand, Cottager's Kale is an ancient variety from England that still throws a proportion of perennial plants.
https://www.quailseeds.com/store/p514/Cottager%27s_Kale.html Western Front is a modern kale bred specifically for homesteaders that also has some seedlings that choose to go perennial.
https://www.quailseeds.com/store/p492/Western_Front_Red_Kale.html
My climate (zone 7) is mild enough to overwinter collards, which are an even richer field for experiment, since there were so many that were grown only by one or two families for their own use. Several of these have been rescued by the Heirloom Collards Project,
https://heirloomcollards.org/ as well as Southern Exposure Seeds, and The Seed Savers Exchange. Old Timey Blue Collards
https://www.quailseeds.com/store/p337/Old-Timey_Blue_Collards.html are perennial for me, and I've seen other (unnamed) varieties around old homesteads.
Chard, arugula, chicory, and celery all have varieties that are closer to the wild and produce a high proportion of perennial plants. Older varieties of leeks do sometimes as well.
https://www.quailseeds.com/store/p9/Perpetual_Spinach_%28Leafbeet%29_Chard.html
https://www.quailseeds.com/store/p295/Perennial_Arugula%2C_Rucola_selvatica%2C_Wild_Rocket_%22Sylvetta.html
https://www.quailseeds.com/store/p472/Chicory_%22Trieste_Sweet%22.html
https://www.quailseeds.com/store/p10/Leaf_Celery.html
There are obvious reasons why people are interested in perennials--it's great to have a crop each year without replanting. Further, perennials often produce in early spring when they are particularly welcome. Rhubarb, artichokes, and asparagus all provide the quality, flavor, and quantity to justify their space. But it's easy to assume a perennial will be better when that may not be the case. I don't get quality collard leaves in the spring and summer for example. The plant is too busy flowering and making seed, and it sucks energy from the leaves to fuel seed production. If I want lots of top-quality leaves, I plant new every year and harvest the first-year vegetative growth. Similarly, market gardeners usually replant salad perennials like sorrel, chicory, erba stella, and miner's lettuce rather continuing to crop and cut the same plants. The younger leaves are milder, juicier, and produced in greater quantity. If it's quality leaves you're after, a young plant is what will produce it quickly and well.
Sterile hybrids like Purple Tree Collards avoid this problem, but there are not many such plants, and they are less hardy than their seed-bearing relatives. Like everything, it's a trade-off. And like everything, there is work that could be done to select for less reproduction and more vegetative growth. In my opinion, and with my situation, the amount of work to prepare many perennials, and the smaller quantities they produce over a shorter season than annuals, makes them an adjunct rather than a staple in my garden. If I were in a different climate, or I preferred scrubbing small roots to taking care of seedlings, the balance would be different.
Self-sowing annuals are a good compromise in many gardens. Mustards, turnip greens, cilantro, and miner's lettuce are all self-sowers in my garden. However, as some people in this thread have noted, mulch or established plants or compaction all can prevent self-sowers from germinating. The fact is that most of our food plants are species that thrive in disturbed soil.
Even the wild miner's lettuce on my land moves around, depending on where it can find bare or disturbed soil that hasn't been taken over by grasses. After two years it usually disappears, having been crowded out by stronger more competitive plants.
Human intervention has always been a necessary condition for abundance of human food--from the intentional burning done by California native peoples, to the intentional planting of native fruit trees in the jungle, to the digging and spreading of root and tuber crops in various cultures, to the creation of a plowed seedbed. Each gardener needs to decide the type of interventions that suit their situation, preferences, physical capabilities, resources, and climate.