The film suggests that the 1955 Doc repaired the time circuits before Marty went back to 1885. The letter from 1985 Doc specified that the parts needed to repair the time circuits would be available by 1947, and Marty indicated that 1955 Doc was fully capable of repairing the time circuits on the DeLorean.
1985 DOC: I set myself up as a blacksmith as a front while I attempted to repair the damage to the time circuits. Unfortunately, this proved impossible, because suitable replacement parts will not be invented until 1947.
MARTY: It's not gonna happen, Doc. After you fix the time circuits and put new tires on the DeLorean, I'm going back to 1885, and I'm bringing you home.
Also, the fact that Marty was able to go back to 1885 is evidence in itself that the time circuits on the DeLorean were repaired. There's nothing in the film that suggests they weren't fully repaired.
As for the flying circuits, 1955 Doc presumably wasn't able to repair those, as, according to Doc's letter, those weren't just shorted out, like the time circuits, but outright destroyed.
1985 DOC: The lightning bolt that hit the DeLorean caused a gigawatt overload which scrambled the time circuits, activated the Flux Capacitor, and sent me back to 1885. The overload shorted out the time circuits and destroyed the flying circuits. Unfortunately, the car will never fly again.
This suggests that the flying circuits were irreparable, and needed to be fully replaced, which likely would've required technology from 2015, or at the very least, the knowledge 1985 Doc gained whilst in that time period.
You've proposed that 1955 Doc could've sent Marty to either 1985 or 2015 in order to fix the flying circuits, but let's think about that for a moment. Who would've fixed them in 1985? The hovercar tech didn't natively exist in that time period, and neither did the 1985 version of Doc, since he'd been sent back to 1885 and died there in the then-current timeline.
Theoretically, Marty could've time travelled to a point in 1985 shortly before the events of the first film and interacted with Doc then, but interacting with Doc at that point would've risked a major disruption to the timeline, and thus a time paradox. It was hardly worth taking that risk in order to fix the flying circuits on the DeLorean, which weren't necessary for what Marty needed to do in 1885.
Also, that version of Doc hadn't been to 2015 yet, and likely didn't have the knowledge required to fix the flying circuits, since he didn't invent that technology himself.
As you proposed, Marty also could've travelled to 2015 with the intention of having the flying circuits fixed, but who would've fixed them there? Again, the older version of Doc who was familiar with that technology had been sent to 1885 and died there in the then-current timeline. You mentioned a "service shop," implying that a random mechanic from 2015 could've done the job, but allowing some random mechanic to work on the DeLorean would've risked that person discovering the time travel technology in the vehicle, which Doc kept closely-guarded for obvious reasons.
We don't strictly know whether 1985 Doc allowed others to work on the DeLorean in order to upgrade it with hovercar tech, or whether he performed those upgrades himself (using knowledge he acquired in 2015). Ideally, I think he would've preferred to have performed those upgrades himself, to minimise the risk of others discovering his time travel tech. But if he did allow others to work on the car, I imagine he likely would've removed at least some of the more conspicuous components of the time travel tech first, to help keep that stuff under wraps.
As far as we know, Marty wouldn't have had the knowledge required to uninstall and then reinstall things like the Flux Capacitor himself, so if he'd have taken the DeLorean into a service shop in 2015, he'd have had to've left that stuff on full display. It doesn't make sense to think that either he or 1955 Doc would've been willing to take that risk, especially in order to fix the flying circuits which, again, weren't necessary for what Marty needed to do in 1885.