We have the same root problem though one is a huge oak that was cut-down before we moved in and the neighbours have huge leylandi… digging is a nightmare. Even where we have dug we found stuff like unexpected concrete paths (1920’s build) that are 2′ below the current lawn.
We have lots of deep planters made from decking… these can be moved about a bit and then some flowers and shrubs in the borders. Perhaps by the time the planters rot the roots will be rotten as well?
Then lawn and patio and some planters on patio… (didn’t want to have to rip up patio and its kinda useful) especially in winter but also for the BBQ etc.
We grow quite a bit of food and most of that is in planters (easier to keep pests off)
Don’t know if that helps but its another way of cultivating if digging is a real pain.