King Seiko 36000bph(45XX) vs 28800bph(5626)

FIFY

I'd say both are equally good quality movements. There are pros and cons of manual wind Vs automatic and likewise 36000 Vs 28800.
Personally, I prefer automatic and if you've found a good example and it's running ok and you like it, then I'd go for that...but everyone's different. Both are good movements, but if you have a particular desire for manual wind or 36000bph then that may sway you.

I don't think that desirability is currently correlated very much with beat-rate, although it was a selling point at the time. Other things make watches collectible. Personally, I would be focusing on grammar of design pieces if I were going to start a vintage KS collection.

Google "seiko grammar of design" and you will learn a lot about collectible vintage Seiko. Watches are more than movements.

Good site on diy servicing vintage Seiko.

That's pretty inexpensive to service a mechanical watch (or maybe they just plop a new movement and call it a day... 😀).

Lots of parts are out of stock in the official seiko system, and they don't produce new ones for obsolete models. Also, at least in Europe, the official seiko service center tends to refuse working on watches on which they lack readily available components...

Older KS are well made, neither easier nor harder to work on than other comparable watches. 562x are tough, I rarely saw worn parts. Lack of spares means you better not fumble unless you don't mind buying watches for scavenging.

The 45xx movement series is super interesting in the history of Seiko - these were the (mechanical) movements that scored highest in the 1968 Neuchatel observatory trials, putting all of Swiss watchmaking behind them. The next year, the trials were shut down ... certainly for other reasons ;-)

I love having a manual wind 36,000 vph. There have been people saying that the Lord Marvel 5740C high-beat movement has an even smoother sweep, but I really don't know how this would work, as they both run at the same frequency. Still, something to keep in mind while exploring.

There is one thing that is seen as a bit of a weak spot with the 45xx movement series: they use a lot of torque to keep the movement running, so there can be a lot of stress on the gear train. It is recommended not to wind them completely up to the max. Personally, I haven't had a problem, but it is something I read on occasion.

Here's a link to a discussion on a 45-8010 service:
https://omegaforums.net/threads/king-seiko-45-8010-chronometer.160997/

You may also find this interesting:
https://www.beyondthedial.com/post/king-seiko-collector-guide-part-2/

About the 45ks series : buy one in good shape or well priced, because the torque can eat the gear train... And you end up buying two to have a working one. Ask me how I know. Not a problem in my experience for the 28800bph 56ks series.

If the watch is running at +- 1s/d it can be an indicator that someone recently serviced it. But it isn't a 100% certainty - a watch can run really well (as in, precise), and still be in dire need of new oil. A better indicator is the amplitude, but these do differ quite a bit between movements (high-beat Seikos tend to have lower amplitudes even serviced).

Yes, the date corrector wheel is a bugbear of the 56xx movement series. There are aftermarket steel replacement parts that you can buy, though.

A single "accuracy" number really doesn't indicate much. At the very least, you would want to measure timekeeping in various positions, amplitude, etc. And also inspect the movement to see if lubricant is present. Even after that, there's no way to know if the movement has been maintained well over the years, and some parts could need replacement.
Having bought hundreds of old watches, I can tell you that it's a crapshoot, there's just no way to know what has happened over 50-80 years of life. Sometimes you get lucky, the movement is in great shape, and all it needs is cleaning and lubrication. Other times you need to hunt down expensive parts. I recently bought a 1960s Seamaster in great cosmetic condition that was keeping excellent time, but it ended up needing new reverser wheels that were hard to find and ultimately cost me $150.

Obviously, you can minimize the risk by sticking with simpler no-date hand-winding watches. Anecdotally, I've found that calendar and auto-winding mechanisms often need repairs and parts.

Also, IMO it's a good idea to buy watches with enough collectible value to merit the repairs that might be needed. Watches that are desirable and in excellent cosmetic condition (clean dial, sharp case, etc). When you or your watchmaker inspect and disassemble a watch, you may easily find that it needs an additional $200 of parts/repairs beyond what you expected, which could put you upside-down on a low-value watch. In practice I find that easier to accept if it's a nice watch. The Seamaster I mentioned above is an example of this, I now have more into it than it's worth, but at least it was worth resurrecting.

It's possible to do a cursory inspection for lubrication by opening the caseback and viewing the movement under magnification. At least you can tell if the oil is totally dried up. But honestly, it's still a crapshoot and I wouldn't rule out dry watches or watches that aren't running well, those can be great watches.

I've purchased many dry and unserviced watches that only needed a simple COA, presumably because they had just been stored in a drawer for many years. As a collector, these can be some of the best watches, because they haven't been tampered with, yet they are still easy to resurrect, even if they aren't running well. I recently bought a non-runner that was in great condition other than a broken mainspring. Most likely the mainspring broke, and the owner just put it away. I took a risk on that one and it worked out. Other times, someone will keep using a watch without servicing it, and the parts develop a lot of wear. Some movements are very robust and will run and keep good time for decades, and owners take the if-it-ain't-broke-don't-fix-it attitude.

The best you can do is to inspect the watch, run some diagnostics, ask about the history (but take the answer with a grain of salt), and use your judgement. You win some and you lose some.

It's impossible to say what is the most likely reason for +40s timekeeping without more information, any original chronometer certification is irrelevant. Unlikely, but it could even be something as minor as magnetization or debris/oil in the hairspring.

TBH, I think you just need to dive in and start learning by doing, there's only so much you can learn by speculation and superficial measures like overall timekeeping. If you're not comfortable with that watch, buy a different one but accept the fact that there are no guarantees. Once you have bought and sold a dozen watches, you will look back on this thread and have a much different perspective.