Your seedlings in vermiculite should be just fine for several more weeks of growth. (At this time of year, that may take more than several weeks...
In any case, I wouldn't mess with them until they have put on some size and made enough roots to hold most of the existing vermiculite together when you tip/tug the plants out. Then all you need to do is add new mix underneath, set the root ball back down, lightly brush away any loose vermiculite from the top layer and add some new mix to cover. The idea is to disturb the roots as little as possible while giving them something new to grow into. Even if that new mix is different, such as bagged potting mix, it won't hurt to leave the original root ball of vermiculite intact. If you're doing this during very favorable weather, such as a stretch of overcast skies with pleasant rains, you could shake the roots gently, mix the original vermiculite with some bagged mix, and use that as your new soil.
Also, whenever transplanting seedlings, don't put too much soil mix over the crowns — 5-6 mm is plenty. (check to see that no roots are showing after the plants are watered.) The bagged mix you're showing in the other photos looks like it could probably benefit from 10% perlite to lighten it.
Most of this is extra-cautious advice, and once you've raised more seedlings you'll have a better idea of what they can take. When I transplant seedlings, I'll sometimes hold a few of the runts back and place them in a jug of water. With adequate sunlight, and the occasional splash of liquid fertilizer, it's not unusual for them to live through the rest of the summer and even through the next winter.
Your garden soil would benefit from a light, but well-maintained covering of mulch. Nothing so fine-textured that it packs down and becomes just another layer of soil, but something that lets air and water through while shading the soil and keeping it moist. Since the native soil seems to be performing well, a mulch is probably better than digging a bunch of organic matter directly into the soil. With a mulch, the soil organisms do all the work for you.
Were you ever able to find alfalfa pellets? (make sure they don't have added salts.) I use them under a coarse mulch sometimes.