For my part, I have never had any problems whatsoever with slow combat sessions or even dice-rolled combat at all. Part of being an effective DM/GM is being able to adapt to the preferences of the player group as a whole.
But I think there's two sides to the argument of maintaining engagement. The first thing I ask myself when I see the phones being pulled out and the increased volume of side-conversations is "is my current style of GM-ing causing my players' eyes to glaze over?" After settling that question, of course, I begin the admonishments, finger-shaking, and scolding.
On a side note, I must confess to experiencing alarm at this rapidly-growing idea that there must be less time spent on combat resolution and more on whatever the group defines as role-playing. For some groups, that just isn't, and will never be, the case.
Over 20 years ago, I joined a newly-formed (and short-lived) gaming group where the genre was 18th-century piracy, trading, and political intrigue in the Caribbean islands. The GM did not inform me that, when he ran gaming sessions, dice-rolling as a means of combat resolution was a thing of the past. I forget exactly how he went about resolving said combat, but I do remember that he treated inevitable fighting as something akin to a visit to the dentist: necessary, and put behind us ASAP. Simply put, the fact being treated as if it didn't exist is that some adult gamers love their dice-rolling combat sessions just as much as they do their opportunities to take the role-playing stage. For them (ahem, us), the opportunity to roll dice is the same as having just a little part in deciding what direction the session takes afterward. Hey, we just slew the villainous overlord and saved the town! That means that, because of us, fate has been altered for this part of the world, for better or worse. That's not altogether an undesirable thing.