Whenever doing mechanical removal, especially with rhizome roots, you're not gonna get them all. Don't focus on that either.
They are like a battery, storing all their energy in the roots for winter and shooting back up during the growing season. If you remove the bulk, you're robbing the plant of a lot of battery, making it less able to pop up.
You'll have it come back. But in smaller amounts, which you can just hand pull as they do, eventually totally starving it.
There are certainly ways to find more roots, or times to pull (like rain or after it grows up) or other methods (like a foliar spray), but these all cost more time or include using poisons in your garden. Instead, just dig it, pull as it crops up, and see how it goes. It will likely be enough for less time and let you get into planting stuff.
Also bear in mind: your garden soil also has a seed bank ready to go, so once spring hits you're gonna get all kind of stuff growing there. If you're planning to grow your own stuff (food plants or native flowers), then you'll want to plant those partially grown after you dig, use a mulch for a year or so, and keep weeding. Eventually your planted plants will establish and inhibit other plant growth.
The mixed lawn is great mostly because it's less maintenance. The clover doesn't really do that much as a flower, it's more the lack of needing to constantly water or use herbicides that make it so beneficial.
Same goes for any non native flower. Yes, generalist bees visit them. It's not helping your ecosystem though. Only plants native to you will benefit the ecosystem. The food source of non natives does not feed all insects nor is as nutritional to the ones who use it. And they can't serve as host for any notable number of beneficial bugs. Instead, they'll displace better native plants, and amplify bad non native bugs (which in turn will further harm native insects)
Obviously a general exception to food plants. Unless it's a known or potentially invasive one (e.g. Bradford pears in southern US cause brutal invasives), you're not going to really get a native food plant, you're growing them for food not for the ecosystem.
While there are less aggressive types of bamboo, the point is it does not belong in your ecosystem. There's better options. Also native plants sited right (light level) will require like no maintenance to keep alive :) It needs help when you plant it esp if you've got a drought, but that's it. Getting a partially grown one from a native store (garden centers don't tend to actually carry natives) is a solid strategy
This finder looked fun: https://buynative.co.uk/plants/
Anyways, overall it sounds like you've been thoughtful about your yard space so you're already doing great ;) send pics!