is there a process or product that works well on skulls that have been in ground contact for many years?

If degreased, you can bleach with the beauty supply paste/kit. If dry you might just get some arts & crafts supply paint. (I did.)
Grafton will know...

BNagel putting me on the spot!

Old stains like that may be hard to remove. I would start with a soak in hot water and dawn dish detergent. You can leave it soaking for as long as you want, I would soak at least a couple of weeks. Take a brush to it and see if the stains/grease are lifting out. Change water if it gets nasty. Let the skull dry and see how white it looks.

If still stained, I would try a soak in acetone or white gas. Be careful with that stuff!

The peroxide paste skull whitening kits would be the next step.

That should help it but they may never get "white"

If it looks good, seal it with a clear matte finish or paint on a couple coats of Elmers glue.

Paint would be a last resort but you can go that route if you want. It is a matter of personal taste but I kind of like a more natural look to a skull vs. a pure white, painted one.

I have had some experience in this. I found a intact deer skull in the woods last year and decided it might look good in my man cave. The first thing I did was to clean it in a solution of warm water mixed with borax. First I soaked it in the hydrogen peroxide you find in most stores, didn't work. Than I went full out and did pure 40 volume hydrogen peroxide. This worked wonders on most of the skull but the nasal cavity was still noticeably stained. i finally ended up hydro dipping it.