Rommel, at least, and probably it was German Army doctrine, routinely used combined-arms battlegroups in North Africa. In 1941, the British were organized along division and brigade lines, but even by the summer of 1941, there was a limited use of motorized infantry, artillery, and anti-tank guns. Including tanks would still have been a rarity.
General Auchinleck and Eric Dorman-Smith advocated the use of combined arms battlegroups, but what were actually created in mid-1942 were generally to small, so that it was not possible to easily concentrate them. Instead, you had these small groups off "swanning" in the desert, all doing what they wanted without central control. That understandably alienated mainstream British army opinion, which was generally conservative and opposed any radical ideas, and the people who advocated them. That was part of the reason that Percy Hobart was unemployed in the late 1930's and why Eric Dorman-Smith ended up that way. Claude Auchinleck had enough prestige that he continued to be employed, at least back in India.