Mint invasion

We have 6 garden beds and one last home owner has this mint in on bed. Now in two years its creeping all over the area that its a jungle out there. We tried clearing two beds last year and dig out the roots as much as we could. Just one bed took us three hours. On one bed we put a tarp and that bed has been but from sides, all mint has come back.

We asked a lawn company to get rid of it. They said we will pull all by hand , take off mulch we have in that area and ass sod in that place(since we asked for it). We are really tired of maintaining this area, we have to spend so much time every month to get rid of weeds n bushes etc. they are asking 2k.
But i feel they just pulling and not digging out the roots, wont the mint be back just next month?

Any other solution we can do? I know digging is best but its too big of a area to shovel. Any tools we can rent/ buy to make digging quicker?