Recommendation for analog set for $700

Good advice from Chayro regarding not to buy vintage. A Rega P3 is a very good start, but only buy from a trusted source such as an original owner or dealer. Don't buy an Ebay special that has been passed around.
If you want to compare tables against the P3, consider a MUSIC HALL or PROJECT DEBUT CARBON as suggested earlier.

For that price range, you could buy a Rega P1 or P2 used, plus a Bellari phonostage used. The Bellari sounds good with the Rega tables. Music Hall is also a decent choice, better than the Project tables.

Pro-ject Debut Carbon, speed box, acrylic platter. Then save your money and get the Ortofon 2M black cartidge and put that on it. Then get some Dynamat and cut it into pieces that fit under the plinth of the turntable. Then if your budget allows, spend some money on some good phono interconnects. I did Morrow PH4, but there are many out there that are very good. Finally, get some Sorbothane grommets and put them between the turntable feet and plinth.

I did this with mine and ohhhhhh yeahhh!!!

I got an impression that the tone arm and cartridge is more important than the table itself, am I right?

I am considering that Debut Carbon also. It has a Carbon fiber arm, is it good or gimmick?

Debut Carbon. Add the Acrylic platter and speed box. If you have some money left, add a ice set of phono cables. I have had great luck with Morrow cables.