Thank you guys for all ideas. I have been really busy with my day job and family for the last few days, until very late at night, when I was so tired that I couldn't even concentrate to write a decent message here.
Was able to have a closer look at this tractor. It seems to me that it sat somewhere and rusted for some years, until some guy got it and decided to put the diesel engine in it and revive it. There are signs that this loader took quite some abuse: some of the holes where pins go are very much out of round. Some linkages bent, some hydraulic hoses pinched. I'll show pics when I get around to it. No cracked frame that I could see, so that's ok. Not so many things missing, as far as I can see: some of the frame parts at the front, for instance the posts that hold the hydraulic tank. All in all, not too bad.
The thing is, I need this tractor functional asap so I can sell the other 644. Funds do not allow to keep both, however nice that would be. Therefore, this time around this tractor will NOT go through a thorough "restoration", rather it will be a "resurrection". Some day in the future, depending on where my life goes, I might have time to do a proper restoration.
Now, the diesel engine re-powering. First, I don't know the condition of this engine. The seller said he never got it to run, he bought the whole package from someone else. Would it be strong enough to use with this loader? I also don't know that. There was no Case 620 that I know of. So, if Case considered that the minimum HP figure for the loader was 14 HP, do I want to risk and put a 10 HP engine in it? Sure the diesel has more torque than a gas engine, at the same horse power, but still, there's quite a distance between 10HP and 14HP.
I tried to start the diesel engine but wasn't able to. Granted I didn't put much time into it. But it has enough unknowns right now that it doesn't exactly qualify for the quickest solution to get the tractor running. Plus, as
@thebuildist correctly identified, there will be complications with the hydraulic oil cooler, and the diesel engine radiator. Another potential complication is the hydraulic pump coupling that the original owner did; it's a little flimsy to me.
I have a Kohler K321 that's been freshly rebuilt, so it should be easier to use for this tractor. I also have a newer 15HP engine with a hydraulic pump properly attached to it, which might fit well in this tractor even though it has no front PTO. This tractor will be only on loader duty, not mowing, so no need for the front PTO. I just have to check what's needed to connect that hydraulic pump to the existing pipes.
@Propane57, I have bought a lottery ticket. If I win over $1M, you will get the running 644 delivered to your door. If you refuse it, it will go to
@thebuildist 
let's go, it's finally time to blow. 