Allium schubertii isn't hardy here in zone 6. Brent and Becky's list it as zone 7. I've planted it before and it died out.
Allium pulchellum has re-seeded quite a lot in a local garden. It might depend on the drainage.
I think Allium azureum produces bulblets in the flower clusters. I'm growing these and love the color. So far I've not thought of them as invasive but sometimes it takes several years to know.

I can't be of much help..as the only allium I have other than Allium Schubertii (which doesn't multiply fast enough) is Allium triquetrum which in zone 8 here is very invasive...but so pretty I keep it anyway. I don't know what it would do in your zone.

Here is a link that might be useful: My exchange page

A. cowanii is a prolific spreader here in Dallas, but I don’t think I would call it invasive. It’s very nice, basically a much bigger version of A. neapolitanium—much bigger leaves, much bigger flowers. The leaves are damaged by late freezes, as with A. neapolitanum. They turn white with a mushy, fleshy texture for a while, then become papery. Freeze damage to the leaves doesn’t affect the flowering. The flowers have a pleasant, sweet scent, but the stems smell oniony when cut. They readily hybridize with A. neapolitanum, with the cowanii features dominating, so if you want both, I would plant them apart from each other. Here they bloom in March and April, and disappear without a trace by late May or June. Nice plant that spreads a little, but is easy to keep under control.

A. sphaerocephalon is very weedy. It reseeds to an extreme degree here in Dallas, zone 8, and must be thinned annually to keep the plants at a showy, ornamental size. Left on their own, you end up with a bunch of overcrowded, spindly, sad plants.

Check out the native wild alliums where you live. Some are rampant weeds, like most Allium canadense strains, A. vineale, and others, although I think it makes my lawn smell like a French bistro when I mow them down. Some are beautiful, like A. drummondii and A. stellatum. In zone 6, you should be able to grow ramps, A. tricoccum, and ramsons, A. ursinum. Both are pretty, and pretty tasty as well. Many culinary varieties of onions, garlic, chives, leeks, etc. have showy flowers if you don’t harvest them for the kitchen before they bloom.