Advice on grouting wall tiles with gaps between wall and tile

I wouldn't be too concerned unless there's mortar that is raised above the level of the grout - which you've already said there isn't.

Scraping the higher sections to give a reasonable depth of grout would be as far as I'd go.

A lot of adhesives these days are sold as tile cement and grout, though they do tend to be coarser than "pure" grout, so they are basically the same material. However, a pure grout does give a smoother, "cleaner" line.

Floor tile grout also tends to be coarser - I suspect to be able to cope with the higher degree of wear and tear it will receive compared to wall tile grout.

In all my jobs, whether large or small tile, using a grout float has always been the way it has been done.

Bagging it would practically do the same thing. The tip of a grout bag is much, much wider than the joints, and you would need to force the grout in anyway, still smearing it over the face of the tile.