This may be an unpopular opinion but White isn't, and has never been, a very good running back. He's not particularly athletic, he's not particularly fast nor quick, he's not particularly strong nor explosive. He's athletically below-average and doesn't offer much dynamism or play-making.
What White is: intelligent, hard-working, a good pass blocker, clean footwork, good attention to detail, good hands. He thrived with Brady given his skill-set, but Harris and Burkhead have clearly been the better RBs in this offense, this season, with Cam at QB. (Heck, even Michel has shown more when he's been on the field this season).
Hopefully Michel can stay healthy and Harris can get over his dings, because White has no business taking carries as far as I'm concerned. Can he be used situationally in spots? Yeah, I guess. But his role should be limited - we simply have better play-makers at RB other than him and his skill-set is less relevant now with the departure of Brady.