You’ll never get “digital quality”. 
That’s not necessarily true. The “weak link” is (usually) the record itself, especially the noise.
Apparently, you are using the same preamp (built-into the turntable) in both cases, and probably neither one has a recording volume control.
The recording level isn’t critical as long as it’s not too loud and clipping (distorting). If one of them is clipping, use the other.
Also, the AV converter might be using some kind of lossy file compression (MPEG, etc.). That’s not necessarily terrible but the “best practice” is to keep everything lossless, and then if you want MP3 (or some other lossy format), convert to MP3 ONCE as the last step.
Izotope has a very good reputation but I’ve never used it.
I have Wave Repair ($30 USD) and it can do a “perfect” job on most (but not all) clicks & pops. It’s manual, which means you have find and select the defects manually and that’s VERY time consuming. But that also means it only “touches” the audio where you identify a defect. It actually seems to work better on the bad clicks & pops ,maybe because they are easier to find. It also has a spectral view mode (to help find the defects), and it has several different repair methods.
I also have Wave Corrector (FREE) which is fully automatic, but probably not any better than what you already have. I don’t have that much experience with it.
Compression makes the low-level noise WORSE, even without make-up gain (because you’re going to listen at the same volume either way).
In general, dynamic compression (unrelated to file compression like MP3) makes the loud parts quieter or the quiet parts louder (boosting the background noise) so it reduces the signal-to-noise ratio.
In practice, usually the loud parts are “pushed down” and then make-up gain might be used to make “everything louder”.
If you have super-loud clicks & pops that are louder than the music, compression (or limiting) it CAN make those (relatively) quieter. (Limiting is a fast-kind of dynamic compression.)
A noise gate is downward EXPANSION (the opposite of compression). It makes the quiet parts quieter, or completely silent. It’s worth trying to reduce the noise during silent or quiet parts but sometimes you hear the background noise going up-and-down it’s worse and more distracting than leaving it alone.
P.S.
I find that lots of older records are a bit “dull sounding” (rolled-off highs) so I’ll sometimes use the Graphic EQ to boost the high frequencies
3-6dB. But compare to a good-sounding CD do you don’t over-do it.
You may find that some other EQ helps too if the recording doesn’t “sound right”.
And at the end, I usually Normalize or Amplify to “maximize” the volume. If you’re used EQ or anything else that might boost the volume, make sure to do that before exporting so that the exported file isn’t clipped. You can normalize the album as a whole to keep the quieter songs relatively quieter and the loud songs relatively louder (as the artist/producer intended) or you can normalize them individually if you prefer.
You may also want to keep an original unprocessed copy in case you later realize that you messed-up something, or made something worse.
