I don't know how much sense it makes to insist on using the turntable's built-in ADC. Chances are it'll be a contemporary consumer-level ADC like CS5341 or AK5359 or similar, run at either 44.1 or 48 kHz... pretty much what you get in $25 A/D converter boxes.
What sort of onboard audio is available? Even a rather basic chip will tend to have a somewhat better ADC in it than a basic USB audio codec (PCM290x), one capable of 24/96 or at least 16/96) recording to boot. By the time you get to an ALC1200 or even ALC1220, you'll be at the level of said A/D converter boxes or even beyond.
In any case, using the onboard line-in via an appropriate RCA to 3.5 mm cable is well worth a shot - we used to record analog media on worse 20 years ago. I would recommend recording at 96 kHz, 16 or 24 bit, setting up both the recording device and software accordingly. (Note: When using Audacity on Windows, 24-bit capture requires choosing the WASAPI audio API. If ASIO is required for a DAW, ASIO4All should do the trick.)
The ST-150 should be set up for LINE output (how good its built-in phonopre might be is anyone's guess), with ground lift slid over to the right (i.e. lifted).
If you want some fancier hardware, I might try a Sound Blaster X-Fi HD... it's a consumer level USB device using a CS5361 ADC that can be connected using ordinary RCA cables and even comes with a built-in phonopre if need be. (It supports 48 or 96 kHz only if memory serves, but that's fine in this application. The line-in also seems to be fixed level, presumably 2 Vrms as Creative cards usually are, but providing digital gain is trivial and unproblematic with 24-bit capture.) Asus Xonar U7 MkII would be another similar option, no phonopre but same ADC (
drivers may be iffy).
While I would recommend recording in 24/96, that does not mean you have to keep things that way afterwards. Even Audacity comes with a high-quality (SoX) resampler, so downsampling to 24/48 or 24/44 is easily doable.
For vinyl recording, about the most relevant performance characteristic is anti-alias filtering. This means the model of ADC used becomes central, next to analog connectivity and levels. It would take something like the iD14 for a substantial step up, and even then the main advantage is being able to record at 48 or 44.1 kHz straight. Oversampled recording will let you get away with a substantially more pedestrian ADC. Also note that newer does not necessarily equal better, as in the last decade or so low latency has generally been prioritized over digital filter performance.