I gardened for many years in Zone 4b/5a at 9/10ths of a mile elevation before moving to Zone 8a last summer. For many of the plants you mentioned, age of the transplant is not a huge issue, as long as while you are growing them out inside you move them to larger pots as their root balls fill the old ones, plus you are careful with the roots and make sure to fill each new pot with a good soil mix so they have every incentive to continue growing well.
In fact, the way we used to get around the short season problem was to start our garden transplants in soil blocks, then move them to progressively larger grow bags as the season progressed until conditions outside were good enough for transplanting. Most years our tomatoes ended up in 2 gallon grow bags (with a plant that was about 16 to 18 inches tall and wide) before being transplanted out in late May/early June. As a result of this practice, we almost always had ripe tomatoes on some of our fastest varieties by mid-July. If there was a setback, you would hardly notice it, and it was more than outweighed by having larger and older transplants to start with. The only issue we encountered with transplants that old and that large was it was a lot more work to get them into the ground!
You could treat peppers and eggplants this way and do fine as well, although they likely wouldn't grow large enough to need a 2 gallon grow bag in the same amount of time, as they grow a fair bit slower than tomatoes do.
Things that would NOT do well with this method... cucumbers, melons and squash. These resent being transplanted much past 3 weeks after germination, so only start those a couple or three weeks before you are confident conditions outside will be warm enough to keep them happy.
By the way, I personally do not like peat pot products. My experience with them has been that they do not disintegrate anywhere fast enough to keep from stunting the root system of whatever is grown in them, even if you follow the standard advice to tear the top edge off, crack the rest of the pot, and then bury it deeply.
If you are interested in trying the soil blocks mentioned above, you can get block makers from Johnny's Select Seeds for about $25.00 US on up, depending on the size block and block maker configuration. I have used them since the late 80s, and am a big fan. They take a bit of practice, but the results I've had were great.