Which digital photo camera would you suggest I buy ?

Either Kodak or Nikon.  I've used both, and wish I still had one of them.

If there was a digital camera out there without video made during that time, it would have to be either exorbitantly expensive or crappily cheap.

If you don't mind a dSLR, your best bet would be one around 6MP. There are still lots around in the secondhand market. Just don't get anything lower than $100, even though how tempting the ad will be.

The Nikon D3 was made a bit earlier than your timeline, but is still a good seller on ebay. Even better is the Nikon D3S for it's improved high iso performance, but it does shoot 720p video. Both are 12.1MP full frame cameras.

Nikon DF. I think there was also a weird Sigma camera.

But why not just not use the video recorder function on normal Canons and Nikons?

Pentax 645D is a great digital camera without any video capabilities. Used, you can get one for around $1,800 body only.  

No Sony, Canon or Nikon made between 2015~2018 is without video capabilities. 

I don't think you'll find anthing that can't record video manufactured so recently - by 2015, that would have made it too difficult to sell.  But why so specific?