In view of all the obstacles confronting you, the easiest solution may be to limit your new garden to species that adapt well to your soil, climate and rainfall. I have no idea what grows well in zone 5, so I'll refer to a similar project I undertook several years ago in zone 8. The challenge was to transform a neglected weed field into something approaching an attractive landscape. The soil was heavy alkaline clay that remained saturated in winter and baked to hard bricks in summer. The solution was the wildflowers and grasses that had evolved under these conditions. I would be remiss if I did not point out that the weed field was such a mess that I spent two years getting the weeds under control before I sowed any seeds. Even then, I still had to contend with weeds (buffalograss doesn't choke them out). If you need faster results, you could apply e.g. shredded bark mulch around your seedlings, which in your situation introduces a new logistical problem.

Short answer: It is possible to grow flowers in unamended clay and without supplemental irrigation.

use more robust species in the first year, and forget about the flowers, or rather do it deviously, using sunflowers and flowering sorghum (not the exact name, but I can find the name if needed). you have to break it first if you want to enjoy it. do not forget to plant a winter cover in fall.

Some plants like clay, coneflowers, rudbeckias, sunflowers, daylillies, joe pye, cup plant etc. Not sure what area you are in. I have never had luck with just sowing seeds from a packet in clay, that's just been my experience (unless it was things like peas, beans and cukes).

There is nothing wrong with clay soils that a lot of organic material will not help.

The compost was a good start. In the fall, when you're done growing, add more compost, then layer compostables over the top as thick as you can. Mix up leaves and grass clippings, or whatever you can find. Let it compost over the winter - a lot will happen in late fall and early spring. The freezing and thawing plus the earthworm activity this will bring, will help fluff it a bit. Let nature do the work.

Avoid walking on it or digging when wet. Make some aisles for walking. Next year either till in the compost layer or plant through it if you're putting in plants instead of seeds.

And use degradable mulches in the summer to keep it from drying out and getting rock hard. By fall the mulch will have decomposed into the soil too.