Since we know that TNG replicators are transporter-based, it is a given that TNG-era transporters use little enough power for that to be practical. The question is whether the same was true in the 2260s specifically. Only evidence from TOS itself is relevant to the question of how advanced the technology was during TOS itself.
Now that presupposes that TNG and TOS transporters are sufficiently different to require separate attention. Why presuppose such a thing, if "technology from the past mustn't inevitably be replaced"? But all right...
Instances where lack of power precluded use of transporters in TOS: none. Loss of primary energy circuits and propulsion in "Court Martial" still allowed Spock to evacuate the luminaries via transporter - or would have, had they not declined. In "Return of the Archons", Landru's attack badly drained the ship's power, but once the attack ended, Scotty was immediately ready to beam up the landing party. Vaal's power-draining antics in "The Apple" were never directly established as a cause in the failure of the transporters, either.
Instances where lack of power shut down other systems in TOS are rare enough, for that matter. "Doomsday Machine" features the "drained" phaser banks, but combat damage is likely to be the root cause of most of the ills aboard the Constellation
Even if the dumbwaiters are limited exclusively to decks 5-8, it's still feasible for there to be one in at least one of the transporter rooms.
All right, that seems acceptable enough. It's still quite a bit of bulk for a system that apparently sees little other use.
...Or does it double as a means of waste removal? Is that how the phaser in "Conscience of the King", dumped in a slot on the inside curve of a saucer corridor, ended up exploding harmlessly, presumably somewhere outside the ship? Too bad the system isn't seen tripling as tube-mail, by which e.g. McCoy could distribute his newest potions to distant locations.
By your standards, they shouldn't use turbolifts to get around the ship, but should use mini-transporters to beam themselves instantly from one room to another. But they don't. They use elevators. Why should it be any different for moving foodstuffs than it is for moving people?
By the conceit that the food doesn't mind if it ends up a bit scrambled. Transporter technology would become more and more prevalent by first seeing limited, niche applications, and then gaining in reliability and repute until cleared for human applications and perhaps even more intricate operations. That's how the "broadcast" transporter is described in ENT, as having only recently been cleared for humanoids - and that's how I'd assume the "cable" transporter to develop as well.
Really, I still find it distasteful to go all steampunk on TOS when the show indeed is set in a fantastic future. Okay, so perhaps there are a few examples of old tech in use - but dumbwaiters sound like a dumb choice, considering they aren't that hot even today. We prefer microwave ovens and minibars, and strive for in situ synthesis machinery of all sorts - technologies that did not hold much promise in the 1960s yet but that could easily be reinterpreted as having been part and parcel of TOS.
I could see dumbwaiters as a backup system in case "cable" transporters fail, but food delivery doesn't sound like something you'd bother to back up (unlike, say, communications where hardwired intercoms would still be very handy). OTOH, I could in general see food production in TOS as multilayered, with replication, protein resequencing, and onboard cultivation of vegetables all playing their part, and with both automated and manual elements to food preparation. Delivery would also be at least three-tier: an efficient automated mode for serving the action stations, a social mode for mess hall dining, and a private mode wherein foods and beverages are delivered by hand. But there would always be room for both high-end and low-end interpretations of the technologies involved.