Thanks all! Really appreciated, and some great pointers.
Just to provide more details for anyone seeing this thread in the future (I've used this forum so many times in the past so hopefully I can pass it on):
We were looking for hatchbacks (Peugot 207, Mazda 2, Renault Clio, Opel Corsa) no older than 2009, with a budget max of around 3500 (ideally somewhere around 2500-3000) with a max of 125,000km, and with green sticker (allowed to drive in cities), Air Con, ABS, within 75km of Berlin.
I ended up going and seeing 7 cars:
3 at dealerships - each felt like cars that had problems. All 3 felt like they'd sat for a long time, with rust around the engine and wheels. One had major problems with the air conditioner fan where it sounded like it would explode (I didn't even bother test driving because the dealer was useless), but the engine sounded a bit irregular as well. One was advertised as private but technically was in a dealership located next to train tracks (felt proper ghetto). Car had suspension and motor issues, and looked like it had been sitting in a barn for 5 years (rust and musty smell everywhere). Last one was Dabag in Oranienberg - lovely dealership, super friendly and professional, but the car had a weird revving problem with the engine. Also, again, rush in areas I associate with a car that's been standing for long periods of time.
Private sellers - One I couldn't test drive as it was unregistered, and was ultimately not in good enough condition to go through the fuss of arranging a red number plate. Two were just super suspicious, one because his story was just all over the shop, the other because he tried to jedi mind trick me with "you don't need to see any paper work". Of these two cars, one was in decent condition but idled a bit heavily. The jedi mind trick car's radio would turn off when you adjusted the air-con.
Anyway, thanks again everyone. Fingers crossed it all works out, and the used car gods smile upon us.
Last private seller car, which we're probably going to get - private seller, friend showed us the car initially. Just because of previous experiences this raised red flags, but friend was nice and also was up front about a suspension issue that needs fixing, and second test drive we met the guy who owns it. Our testing process went as follows:
General impressions: Engine runs well, no signs of rust, steering true, interior well taken care of, new brakes on front, no signs of airbag deployment, bodywork in good condition, seals generally look good, tyre wear even. Check paperwork - service history was good from 2009-2016, after no log book services, but some receipts for different services done. Not perfect, but passable given the price range we're up for.
Confirmation: DEKRA used car buyers check (83 Euros) - Confirmed the suspension issue, also brought up an exhaust problem where the muffler will eventually have to be replaced (but probably not for another service). Currently waiting on quote for suspension and shock absorber repair. If the repair costs are within our budget, or if the seller is happy to reduce his price so the price falls in our budget, we'll buy.
Also ran VIN through https://www.carvertical.com/de
Expand your search criteria to include Damaged cars, and don't include restrictions like "non-smoker's car only". We found a few cars too late which had "boomered" their ad - saying it was damaged because there was a small scrape on the bumper. Also, the searches will omit cars from the search because they didn't indicate something rather if they did indicate it. So if a seller writes in their description "non smokers car" but don't tick the "non smokers car" box, it won't show up in your search.
Of course this means a more labour intensive process, but well worth it.
Also, just from our small sample size, if you send a seller a message with something like "hi, is your car available still? If so could we arrange an inspection?" and they reply something like a disinterested "yes", don't waste your time. For us this was a pretty good heuristic to avoid time wasting scammers. Real sellers are motivated to arrange ways for you to buy their car.