Others may have a better way, but for me, given your listed tractor, if you have teeth on the front bucket, I'd be tempted to try and put them just under the soil and drive forward to uproot as many clumps as I could. If not, I'd basically do the same thing with the backhoe.
Smooth out the ground best you can after, with either the front bucket or even a hand rake (landscape rake, back blade, or box blade if you have access and a machine with a 3-point hitch to pull one). If you have time, wait for disturbed seeds to re-sprout, then spray with roundup or equivalent.
If you don't have the time to wait, go ahead and put down the gravel and then spray the weeds whenever they show up later start to grow through it--that might be never.
I had a gentleman I met one time tell me he had a dirt parking lot with no weeds, it was hard packed well enough that they just didn't grow. Someone told him he should really put down gravel, so he did. The open spaces in the coarse gravel held enough water that it then grew weeds all over and he really regretted having done it.
My thought is remove the grass first, and see if the dirt will support what you want to do as is...I'd only add gravel if there are problems...just my 2c, not knowing what typical soils are like in your area.
For just me, I am not sure the grass is really a problem. I'd just try mowing it first to knock it down and chop it up and see what happens. It could be keeping everything from washing out. But since you have already said you want to remove it, that's how I'd go about it.