There are lots of results provable in this context. In the book I wrote with Heiner Kamps (which is easily found via Google so I won't advertise here!) we looked at the problem of what results in homotopy theory could be proved with a restricted set of fillers for boxes in a cubical enrichment of a category. This applies to your question since groupoid enriched categories give rise to such cubical homotopy theories very easily.
There is an old paper: P. H. H. Fantham and E. J. Moore, Groupoid enriched categories and homotopy theory, Canad. J. Math., 35, (1983), 385 – 416, which also examines this question and of course, some of the classical book by Gabriel and Zisman is devoted to developing GE-categories in your sense.
As Noah points out, these 2-categories are nowadays more often called (strict) (2,1)-categories although that term (without the `strict') also is used for bicategories in which the homs are groupoids. Try looking up locally groupoidal 2-category in the nLab for more on that side of things.