Hire a bulldozer.
Push the debris into a pile, have a load or two of manure (chicken is great, horse may be easier to find and also works well) dumped over the pile. Wait two or three years, enjoy your pile of topsoil/compost.
If it were my job, I'd cut down the hedge to about 1 foot tall, get a mini-excavator and dig up the roots. This will also help loosen the soil for replanting. You could work other areas of the soil as well.
Disposing of the debris is a matter of choice. I like Ecnerwal's suggestion if you have the space and patience. Or rent a chipper and create chips ...
One piece of advice - most 'bushes' i.e. trees, shrubs and weeds just start growing again if you chop them down, and as its spring, they'll do it pretty soon after cutting down. Most conifers do not regrow once cut down, but most other plants do, so I'd suggest only cutting down those you intend to remove the roots of in fairly short order, rather than ...
What I did was get a truck with a hydraulic arm on the back and pulled each one vertically out of the ground.
Each Bush was secured by a chain wrapped around the trunk, then up she went.
Roots mostly pulled out, and some broke off.
Fast and efficient with no need to trim or saw.
Obviously any machine with a hydraulic arm will do the job. I just ...
Kissing the ground monthly with carefully calibrated amounts of herbicide can kill off the weeds and leave the bushes standing. As they say, poison is all about dosage. An expert on herbicides, if told your desirable and undesirable plant species, may be able to help.
For instance 2,4-D goes after broadleaf but leaves grasses alone, and is quite safe (...
Chain saw, hedge trimmers, wood chipper or match or online advertisement.
Assuming that you are familiar with using a chain saw enough to be safe. Note that you can forgo the hedge trimmers if your chain saw skills are high enough.
Use the hedge trimmers to come in waist to knee height to locate and get access to the main truck/root. Then use the chain ...
A 4x4 pickup with a low-range transfer case should help. Wrap chains (not straps, which can snap violently) around the trunks and creep the vehicle to pull them out. The larger ones you may have to treat more like trees and grind the stumps.