Best TT under $1k

If you can find a nice clean Rega like a early P3 for a good price like $450 or so, then you can get your foot in the door.
 Then you can shop all the various upgrades that are out there.
Each upgrade can make for a nice improvement and can make it pretty interesting and fun at the same time !

+1 chackster, OP not going to get all the priorities listed for less than a grand. I agree with chackster a used vintage DD in good shape will perform better than a cheap BD new. I still use my vintage 76 Sony PS4300 DD to compare with my VPI and although 40 years old it still satisfies. Both get playtime.

I wouldn’t let the inclusion of a cartidge dictate your purchase. If you want a very good NIB table with warranty, the Pioneer PLX-1000 is the best value by a country mile. Order it from somewhere like Needle Doctor or Music Direct and ask them to install a cartridge like the https://www.needledoctor.com/Audio-Technica-VM540ML-Phono-Cartridge

Keep in mind that your analog system will need a phono preamp unless your receiver or integrated (or separate line stage preamp) already has one built-in. I’d go with the Mo-Fi Studio Phono at $250. This would put you a little over budget at MSR prices, but most dealers will gladly grant a 10% discount simply for asking. Tell Needle Doctor you’d like the whole package for $1K. That would yield a very respectable starter system IMO and provide a path to support should you run into any problems.

There is a Denon DP6000 for sale $790 on another site related to an audio asylum.

You can do quite well in the used market for your price range. As already recommended, a vintage VPI (HW-19 or even an HW-19 Jr.) will serve you well. There are many other options that will serve you as well as (or better than) the options you've listed.