Well, there is no actual 'Best', only subjective opinions of what is best. In your situation and price range I would look for a Resolution Audio Opus 21.
The Audio Research CD-3 MkII is close to your budget and is an amazing machine. A way to get a 'world class' machine is to find an EAD Ultradisc 2000 for about a grand and then send it to Noble Eletronics for SOTA mods for another grand. For $2,000.00 plus your time and trouble you'll have a machine that will holds it's own against anything in the market.
with that budget you have a wide range of players to choose from. But to pick the absolute best player, impossible. I can say their are audiogon members that will tell you including myself. That the old JVC 1050tn player at a price of 150.00 used or an older Jolida tube will sound very competitive compared to newer, and will keep much needed cash in your pocket for plenty of used and new Cd's. This is just my take on newer gear, and many will disagree.
The Bryston BCD-1 is worth considering and in your price range if you consider demos or the dealer is willing to deal. Check out the reviews. It put out impressive and involving music.
From my personal experience, I believe a used Modwright Sony 999ES Platinum edition with tube power supply to be exceptionally good for $1000 - $1500. It's very close to my $6500 APL Denon 3910, (which has bested several +$10,000 players in a couple comparisons).
I recommend the Modwright highly (and superb customer support too).
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