Sure . . . for my experience but I may not have made it clear . . .
Operator/Crew level handles the daily stuff and is carried out by the driver/crew of a armored vehicle. For example, as the driver I was the one responsible and signing off on stuff but I was usually overseeing the FNG as he learned how to do it. According to US Army TO&E the driver is supposed to be the newest guy as the part of starting to learn the job, in practice . . . well, driving allows you to see and know what is going on rather than rattling around in the dark of the vehicle so people with rank drove but lowbies did most of the grunt work. If it was too big a job, or needed to be done ASAP the crew assigned to the vehicle jumped in when told (hey, they just told us we need to PMCS that humvee over there before anyone can go to lunch- its not going to be 1 guy doing it, the whole crew will get on it).
For BT mech purposes, this will be the mechwarrior and his personal tech (at least before tech team rules) who is the one personally responsible for the machine and signs off on everything. The mech captain or crew chief (pick your term) may grab astechs out of the general labor pool if needed.
Support level depends a bit on doctrine . . . when I got in the battery/company had a squad of mechanics who were experienced with all the vehicles, controlled the parts & POL, had the more expansive tool chest, and drove the wrecker/recovery vehicles. Each vehicle was issued a 'pioneer kit' and a specific tool box, but if you needed something specialized they were the ones who controlled it. They also checked your paperwork and either verified by complete inspection that you did what you signed off on (no shammin'), spot checked, or took the papers b/c they knew you knew your excrement. They fell under the commander's authority but they worked a lot with the next echelon's support element.
Which was a battalion level support element, who had more general mechanics along with a few specialists. If we had a electronics failure, BN had a specialist who was responsible for software (sort of) and absolutely ensuring hardware was set up right. The one time I dealt with this person, she had soldering equipment and all the other stuff you would expect from some electronics workshop in the 80s or 70s. BN had more wreckers, and at one point was expected to set up a field garage . . . not sure if it was a tent or inflatable structure. It was great for showing a movie on the side of the structure during a down period we were in the field. They were under the command of a Warrant Officer.
During the Brigade Combat Team re-organization some support elements were consolidated at the battalion level due to a doctrinal change. Before the BCT shuffle, and after, Brigade also had a level of support but I never heard of a vehicle turned over to them for care or requiring one of their techs to come down to battery or battalion level, so they may have just been to take care of the BDE HQ assets. IMO, you could look at it as in 3025 when the basic garrison deployment to a planet might be a company, it would make sense for the company to have a slice of the battaltion's tech support while the BN HQ & assigned mech company would have the BN level tech support as part of HQ. When we get to 3050s with battalion garrisons being the norm I think you will find them more consolidated unless the companies have to be spread out with little way of connecting.
Depot level support is going to be regional as I mentioned, so things like PDZs, theaters, or whatever unit of measure. I doubt depot level would be assigned to a single regiment unless it was the regional primary defense force- like a national mech regiment being able to take charge of planetary militias, so it would make sense to co-locate it with the regimental HQ position. To expand on my earlier example, lets look again at the Capellan March . . . so by strategic doctrinal organization the book might say that each of the seven PDZs has a support depot set up, or if they do not want that many as targets the FS/FC might have two in the Cap March, one for the Kathil AO and one for the Taygeta AO. Depot level support is likely under regional command, depending on what that region size is to begin with . . . though I guess you could say the March HQ would be in charge of the PDZ Depots rather than PDZ commanders.
'Factory' level support that will not be based out of a operational factory might be the DS mobile operation like you suggested, especially if there is a big refit program (like hey, 3051! or 2SW tech-downs) operation underway. Place that 'Factory' level team down in a central protected location and instead of it taking months for that company of Vedettes to travel somewhere, be refit with UAC/5s & other upgrdes, and then months to travel back . . . instead the 'Factory' comes to a region where everyone ships their Vedettes for a month of travel. IMO this sort of level of support would be under the Army command structure, so for our Cap March example of a Kathil (b/c producing factory) probably gets it assignments from the Quartermaster Corp HQ on New Avalon.
Does that help?