There are managers who do not do coaching and involved very little in tactics. One of which is Allegri. He's got Aldo Dolcetti doing tactics for him. SAF himself called himself a tactical dinosaur and started relying on others to do the training for him. In fact his spat with Keane started because he hated Carlos Q's drills whom SAF himself described as effective but a bit repetitive. There are other aspects that managers do that are as important as coaching, one of which is man management. With so many prima donnas all of which protected by super agents its so easy to lose the dressing room. Then there are other aspects of the game like selling his vision to the board, making sure that every component of the team are on the same page (youths, data analysts, scouts, coaches, DOF, fitness coaches, doctors etc) and analyzing the tons upon tons of data each segment provides him. I don't know what's your line of work is but I would have to sweat blood to persuade my boss to go for an investment that would cost millions which is exactly what managers do when asking for this or that new player.
Call me old fashioned but I like the manager to be able to do everything. The reason being that understudies might leave or might get complacent which in turn requires the manager to immediately step in and correct things. He also need to be able to identify if his coaches are doing a good job or not. A manager whose tactically inept or can't run a training session would never be able to do that. I believe that one of Ole's biggest mistakes was that he trusted his coaches blindly. That was clearly evident when he left for a holiday in Norway in the middle of a crisis. Which is hilarious really especially considering that the same coaching staff were around during Mou's final disastrous year.