I'm in a pretty similar area to you and have been pondering this same question for the past 5 years. I decided that I wanted something that in addition to suppressing weeds, was either useful (food, flowers, nitrogen) or was a native plant (because "native" tends to be shorthand for exceptional wildlife value + low maintenance). Here are some things I've tried:
Pussytoes & skullcaps (full sun): native, kinda cute, good in hot dry sun. But they don't spread very fast and I am not sure about their weed-suppressing abilities.
Creeping veronica (full sun): spreads nicely, a lovely mat of flowers for a brief period, handles foot traffic, seems to be fighting off competition very well in my terrible weedy boulevard. Not especially useful other than the flowers.
Creeping thyme (full sun): There are different varieties and I think the weed-fighting abilities will vary. Nice flowers. It's edible, but will you reach for it when you have an upright thyme plant nearby that is easier to harvest and not dirty? Hasn't been 100% hardy for me.
Violets (any): I like these, because they are pretty, native with great wildlife value, edible if you care to eat them, tolerate tons of foot traffic, and show up voluntarily. I've been encouraging them but don't have them massed in any one area yet. I'm guessing they won't be great at out-competing weeds.
Canada anemone (full/part sun): I sourced this one because I was looking for more native groundcovers. It's a bit taller so not for foot traffic areas, but spreads well and nice flowers.
Poppy mallow (full/part sun): Same idea as Canada anemone, but this one hasn't spread at all for me. Pretty, not much of a groundcover.
Dutch white clover (any?): A classic, and all-around good at what it does. Not going to overcome established weeds, but good if you let it get established first.
New Zealand giant clover (any): Taller and very lush. I have been thinking this would do a good job but haven't given it a proper trial yet.
Yarrow (full/part sun): A bit taller, but can be very dense. Pretty. I'm not sure if this is a native here or not.
Dandelions (any): Not like I sought them out, but you know. Unfortunately, even if you let them get huge and dense and happy, they still don't keep out other weeds very well.
Partidge pea (full/part sun): Maybe technically the herbaceous layer, but I think it deserves a nod for being a vigorous annual that reseeds readily, is native, is gorgeous and hugely popular with bees, and fixes nitrogen. I have a big stand of this that has been expanding each year and thoroughly keeping everything else at bay.
On strawberries, the different species and common names get confusing, so it's worth asking which kind you've tried. Fragaria x ananassa is the cultivated one—I haven't tried this in a naturalized setting. Fragaria vesca, "alpine strawberry", is the one I most often see billed as "wild strawberry". It has small sweet berries, doesn't spread much (or at all?) by runners, and generally behaves itself. Nice plant, not a great weed suppressor. Fragaria virginiana is the one that's native around here. Also small sweet fruits, but spreads by runners like mad. I planted six plugs a couple years ago and they have formed a huge solid 10'x30' carpet of plants. There's a few shoots of quack grass making a go of it, but the strawberries are winning. If you're looking for a proper groundcover, Fragaria virginiana seems like the clear way to go.