Will a $700 turntable outperform a CD player?

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.

If it has the right phono stage!

To OP, 

There are many variables involved.  If you have a well mastered CD and compare it to a poorly mastered LP or the LP is very noisy, then CD can be way better sounding.  Vice-versa too.  

It really depends on your determination to source and procure well mastered versions of albums on LP.  Things like where and who pressed the records also comes into play.  You might get frustrated being a few new records only to find that they are damaged right out of the package and need to return and get replacements.  Or maybe you want to seek out the very best pressing or mastering of a particular favorite.  If this is your cup of tee, then you might want to get into the vinyl game. 

If so, I'd recommend getting something like a Project Debut Carbon turntable with an Ortofon 2M Red cartridge pre-installed for you.  From a reputable dealer that will accept returns.  This way you might find that it is not for you and you can either return or sell.  Either way, your initial investment is not that high...I think $500 or less.  Especially if you already have a phono input in your receiver or amp to try.  If not, you will also need to get an external phono-pre-amp (aka phonostage).  

Otherwise, you have a good DAC and I think it might be better to explore getting a streamer.  The Bluesound Node 2i is a great product for not a lot of money and can introduce you to the world of music streaming services such as Tidal and Qobuz.  This is the best way to explore new music!

It also depends on what you listen to.  If most of the music you like is newer and mastered on digital, then I think there is little benefit to getting a analogue vinyl front-end.  I find that albums recorded in the analogue domain such as Blue Note albums from the 50's and 60's and remastered by Kevin Gray and pressed by RTI sound magical!  Much better than any digital files or CD.  However, Whitesnake (1987) sounds just as good (or bad LOL) on the CD as the LP.

Lastly, generally speaking a well mastered and pressed LP can be more enjoyable to listen to for most people.  After all we hear in analogue.  My experience is that a decent $400-500 turntable can sound great.  

I have a Technics SL-1200 family turntable and I absolutely love it. But CDs still sound better for the same money invested in hardware. Okay clearly that’s debatable. What’s better, potato salad or coleslaw? I like them both, but the main reason I still have a turntable is a matter of what I can get in that format. Old records of obscure things that aren’t available streaming, and my existing collection of vinyl is what keeps my turntable connected to the "good system" in the house. If I were only streaming and wanted to start a new media collection that I could keep on site, I’d be going for CD - no question. The reason is simple. CDs today are where LPs were in the mid 90s - abundant and very cheap in the used market. And - added bonus - used CDs almost always sound exactly the same as new CDs!! That really isn’t true with vinyl. If this were 1994 I’d be telling you to get a turntable and take advantage of all of the wonderful records that sell for pennies, but those days are gone. As for absolute sound quality, again it’s not an apples to apples comparison. Vinyl has certain qualities that CDs do not, and vice versa. On paper, CD smokes vinyl. But that’s like saying lobster smokes Twinkies - not if you want something sweet it doesn’t.

How far up the food chain do you go for X$ in CD vs vinyl? In my opinion you get way closer to the top performance in CD than you do in vinyl. Because vinyl is a mechanical reproduction system (doesn’t sound as elegant when you put it that way, does it?) there’s almost no limit to what you could do to improve it, and that means cost. CD being and optical/electrical/logical system can only be pushed so far before there is no meaningful upgrade. I think you get very close to perfect, in terms of true sound quality, for much lower cost with CD. That was always been CD's fidelity advantage over vinyl - cost, not absolute quality.

mtdining

... your brain misses the 80% of timing info that digital throws away.

Please explain what leads you to believe that digital discards timing information.