There were plenty of privates (or, in Red Army terms, Red Armymen) in the RKKA. Tables of Organization and Equipment show privates in the same roles you would expect them to fill in Western armies: rank and file soldiers and unskilled or low skilled labour. For instance, in the tables issued on April 5th, 1941 (TO&E 04/400-416) a rifle battalion had 36 officers in high command posts (battalion commander, company commanders, platoon commanders), 102 NCOs (referred to as junior commanders) filling in either technically skilled roles (telephone, radio) or smaller scale command roles (squad commander, gun commander, MG crew commander), and 640 privates, chiefly riflemen or submachinegunners, but there were some in communications, medicine, gun crews, and even scribes.

About a year later, there is not much of a change. A rifle battalion contains 37 officers, 81 NCOs, and 591 privates. The majority of the heavy lifting is still done by people without any kind of fancy titles. You mentioned bread bakers: the field bread bakery has 4 officers, 6 NCOs, and 86 privates. The medical unit is pretty officer heavy, but even here the privates outnumber the rest: 30 officers, 19 NCOs, 54 privates. In the veterinary unit there are 3 officers and 8 privates.

Unfortunately after that I don't have any more TO&Es for infantry battalions or regiments, but a table for a penalty rifle company in 1944 is still as private heavy as always. There are only 12 officers and NCOs and 131 privates. The scribe now does double duty as an armourer, and is a Senior Sergeant, but the second armourer is a private. The senior cook is a Senior Sergeant, but the rank and file kitchen staff are privates. Nurses and horse riders/cart drivers are privates as well.

The veterinary unit structure does lose two privates in 1944, for a total of 3 officers and 6 privates. 3 of them are in the medical section (no duties specified), the other three are in the transport section. One of them is the horseshoer and driver, the duties of the other two are not specified either.

So to answer your question, while there were definitely NCOs in rear line duties such as bread baking and clothes washing, the bulk of the manual labour was done by mere privates. The NCO (or sometimes even commissioned officer) would have been there in a supervisory position that actually entailed some degree of command.

Of course, in specialist units, you would be less likely to find rank and file privates. For example, in order to increase the prestige and training of tank crewmen, the bar for ranks was raised. The lowest rank you could be to serve in a tank crew (on paper) was Jr. Sergeant, serving as the machinegunner. A machinegunner in an infantry platoon would still be a private. You do see exceptions, however, especially towards the end of the war. It would not be unheard of to pick up a private as a loader.

Sources:

TO&E 04/400-416 issued on April 5th, 1941

Central Archives of the Ministry of Defense of the Russian Federation fond 345 opis 5487 delo 366-1 list 405-406

Central Archives of the Ministry of Defense of the Russian Federation fond 214 opis 1437 delo 169 list 60-63, Wartime TO&E of a Rifle Division (unnumbered) April 14th, 1942

http://tankarchives.blogspot.com/2018/06/ranks-and-posts.html

https://tankarchives.blogspot.com/2019/06/big-caliber.html

https://tankarchives.blogspot.com/2018/10/penalty-company-inventory.html