You shouldn’t need a professional and definitely not a digger to remove these.
The buddleias cut the branches down to the ground, if you don’t want to keep them after this, just dig them out, but cutting down is easy, they do grow back fast though. So cut down then remove root ball.
The sedge grasses just use a fork, fork all the way round it, and then lift it out. You can cut it down first, I have even used a bush cutter on one of mine. You certainly don’t need a digger to remove them though and I’d be concerned about anyone who told you you did.
I have about a dozen to eighteen of these sedge grasses, most self seeded, but I’ve a large garden, we recently moved several of them to bare bits in borders and put a couple in large pots. I physically couldn’t move them, but my husband did it relatively easily. He just forked round and lifted. They weigh a lot when you get them out, but look good in big pots.
It’s certainly not a digger job. I’d look for a second opinion. Someone is ripping you off,