Will a $700 turntable outperform a CD player?

MAYBE depends on the cd player if it is an Esoteric K03XS then yes.

Yes.

Mission 775S turntable, $100-200
Grace G707 tonearm, $100-150

Phonostage, $250-1500
Cartridge, $100-10,000

Any of that, in good condition, set up well, will outperform CD

Maybe,

But in the long run, do you want to pay for the maintenance of a turntable.
Turntables are great from the retailers point-of-view. Because they wear out. Which means they have a new customer and additional cash flow over time from this sale.
New belts, new cartridges, bearings replaced in the tonearms, possible motor issues and then the suspensions and occasional tuning to make sure it is up to specs. Plus, all the accessories to upgrade and maintain your TT and record collection. (More so with belt drive TT than direct drive TT)

If your buying a TT, make sure it is new and not used. A used TT is like buying a 20 year old BMW and thinking it will perform just like a new car. 

When a Paul McGowan of PS audio states that DSD audio sounds better than vinyl, I think you can see where the future of audio is going.

Finding LP's at great prices is fun, that's why I still have my TT and to listen to my collection of Lp's. With the prices of used CD's declining, maybe the bargains will be this format for awhile. 

The easiest way for you to resolve this issue is to A-B the same recording on a turntable and in a CD player. I did this years ago, not only for myself but for many friends and family members.   Everyone was amazed as to how much better vinyl sounded than CD.  The 
CD sounded as though it was muffled compared to the vinyl. I urge you to conduct this simple but revealing test. 

No DJ table, or direct drive table will match an “audiophile”turntable designed in the post mass market vinyl era. I would consider the Clearaudio Concept the most affordable destination turntable. The closest thing to a Concept under $1k, would be a Project X1.
This will sound more natural than digital in some ways. Pretty much anyone can hear the difference and is drawn to vinyl IF they are not focused on low level noise, pops, etc.
The above turntables are quieter than you would expect.
i get to listen to the best DAC in the world - the DCS Vivaldi, anytime I want at work. Even that can’t quite match a good vinyl rig for naturalness - your brain misses the 80% of timing info that digital throws away.