If you look at most modern meters, you'll see that the 'rotary switch' is mostly part of the main PCB. It has a carrier with contact(s), and the switch detents are molded into the case itself, with a plastic spring detent mechanism. Minimal parts count for a very functional switch that is both east to actuate and at the same time protected fairly well from accidental movement. It physically switches connections, and they then stay connected. The pushbuttons on the top row of such meters are momentary, and activate logic level functions - they don't/cant do switching like the purely mechanical rotary switch can - you'd need them to turn on/off internal circuits that could then do the switching, and likely with more loss than a purely mechanical contact that's just there, not to mention they'd draw power (not good in a handheld, battery powered device) while the mechanical switch's contacts are purely passive in that regard.
If you look at the pushbuttons on bench meters, they're either logic level and drive power consuming components, or are mechanical, in which case they are likely considerably more complex than the simple rotary switch currently used. Look at the early Fluke handhelds that had the buttons along the left side - those are a mechanical nightmare compared to the low parts count rotary currently in use - they each consist of many components (individual contacts staked into a base, a slider, moving contacts in the slider, a spring to force the slider to its 'home' position, a mechanism to lock the slider in the desired location, for ganged switches some means to interconnect them to each other to disengage one when another is pushed.
Look at these from a Fluke 8200 bench meter, for instance:

Before even adding the keycaps and the stuff to hold them together and couple the mechanism, I can count AT LEAST 12 parts in EACH switch - from the front, the plunger, the spring retainer, the spring, the switch body, six fixed contacts and two moving ones. There may well be more, too - the plunger may fit into an internal carrier for the moving contacts, and they may be spring loaded themselves. It's a pretty complex little piece of hardware. now add six more of them and interconnect them to work properly in unison - more fun.
In short, manufacturers largely stick with the rotary switch because it's simple and it works.
-Pat