I have a skid steer with a grapple, an excavator and thumb is my primary tool and a track loader, I seldom take a dozer to a grubbing job till everything is gone and only the leveling is left to do.
As has been stated check fire codes, here we can burn it if its in a pile, to burn like you mentioned is illegal, its classified as setting a forest fire and can be thrown in jail, but its legal to burn off grass with a permit, don't ask me, I've given up trying to get an explanation from anyone that makes sense. But I will say this much, to pile it up does a better job of burning the stuff and you won't have odd pieces to go pick up later in the ash, in piles it'll burn hotter and quicker and burn more completely than having it lie around like it is now. I don't like to do clean up after a fire like that, it helps to have a rain on it to settle the ash but its still a total pain in my opinion to slop around in the ash later on.
I'd either use an excavator and thumb to pick the debris up, put it in a pile and burn it or use a skid steer and grapple to do that, then use the excavator to dig out the stumps and pile them up too and burn them, I've never had a problem burning any stumps and I'd forget digging a hole to bury anything, if its dry use the debris to get the stumps to burn and use the bucket on the loader if its got teeth on it to back drag the site and your done. When to jobs done and everything is burned up, we flip the fans around on my dozers or crawler loaders and spread the ash piles out over the site, but do it a way to keep yourself out of the ash as much as possible then take the machines home to wash off. I seldom bury anything on a job like you mentioned, mainly because I'm the one to tile the land later on or do the landscaping for developing it or something somewhere on the property eventually and it never fails to have to do something where the debris was buried and re bury the junk again which isn't a fun job by any means.
As for burying it, I'd have at least six feet of dirt cover over it, not just three feet but that's me and I've never dug up a pile that was buried deep enough. We also try to bury the junk along the property boundary seem most area's have a set back law along property boundary's so we dig a line along them and bury the debris there.