Good choice on the 523i. I had a number of happy years with a pair of them and accumulated 150k miles between them. One went to a friend after me and the other is still with my brother.

I'd agree with the idea of giving the cooling system a good going over. However, if the thermostat seems to be behaving itself and there are no obvious signs of neglect, bulging radiator, noisy water pump, nasty looking coolant, cracks or leaks in hoses, etc. then, if you need to do things in stages, focus on a coolant fluid refresh and the other service items and come back to the rest later.

Other things that a car at that mileage will want watching for with be noisy wheel bearings (that can also contribute to vibrations, steering wobbles, etc. as well as noise) and bushes, ball joints, track control arms, wishbones, etc. There are a lot of these and they are wear and tear items and can have a significant impact on the quality of the experience.

As mine got older, the first one I had from 6 years old to 13, the more niggling small electrical issues became a bit of a bother. Nothing major, but in my case with this car it suffered things like the Light Control Module (LCM) issue where one headlight stayed on even when ignition was off and key removed, a failed electric boot release (the bit that whirrs and clicks when you squeeze the boot release on the lid, etc. Pixels dying on the dash is also fairly common but I didn't suffer that with any of mine. None of these were killer issues, all relatively cheap to repair, but they were niggling and caused a few days off the road, etc. At the time I had family commitments that meant I couldn't afford the downtime so it was the only reason I disposed of that car.