Path of signal from turn table to speaker

In a nutshell: no and no.
Its pretty straightforward ( i wrote "simple" and caught myself). With a few exceptions, and those are recent, the signal path is entirely analog. DACs and ADCs are only used for digital sources - and in 99%+ of cases those digital sources are converted to analog for most processing and amplification. That's the function of a DAC and every CD players contains a DAC.

If you begin with vinyl it goes into a filter that mirrors the LP recording filter (an RIAA playback filter) which is packaged with a TON of gain to bring the signal from 1-5mV to roughly 1V. Then to a preamp with very modest gain but lots of switching and controls. Then to a power amp that takes it from ~ 1V up to whatever rated output is - typically 20-40V. The power amp also adds a high current buffer stage capable of driving 2 ohm to 16 ohm loads and therefore delivering several amperes.

End. Unless you are streaming or playing a CD.

Yes, digital amps and processing are gaining ground, but that is very recent and still very rare. Class D (digital) amps are another complication not worth getting into now.

If you are among the 0.1% with a high end digital preamp and amp, then yes you would use an ADC and would essentially defeat the desired characteristics of vinyl.

Moving magnet cartridges need a phono amp with a gain factor of about 100X moving coils need about 1000X. I am keeping this very simple. Both need a filter that rolls off the high frequencies according to a long-established standard. At 1 kHz this requires yet another ~7X gain, assuming it is performed passively.
G

thinking about what you have heard, i’ll add this: in most studios things are immediately digitized and processed in the digital domain, using both digital mixing boards and more often ProTools software.

In the studio case, music -- often individual instrument tracks -- are typically sampled at 24/96, providing lots of headroom to process things without compromising fidelity (in theory). It also allows recording engineers to be sloppy since 16/44 only delivers 96 dB if level sets are done accurately, an art that seems to have been lost around 1968.
I, in fact, believe that it is in the studio where the benefits of HD are clear -- or clearest.  Really well-done 16/44 can be pretty terrific - but that is often shown by 1960s-era verve, mercury or other recordings done simply on tube-powered analog tape machines by loving engineers (like Bob Fine).

Class D is a switching amp, NOT digital. Don't be misled by zealots.
It was invented in 1950.
I've got an A/B and a D and the D sounds better. It's all in the implementation.
There are Class A's that sound shitty with most speakers.
BFD

Guess you chose handle fuzztone because your system sounds fuzzy?

Yes silk purses can be make out of sow's ears but they still smell of pig/hog.

Class A is the gold standard.  It is now largely not offered because of the perceived need to pacify climate change zealots.

Class A/B is a compromise where otherwise competent amp designers have convinced themselves they can have their cake and eat it.  More likely, they know they're pretending.

Class D.  D is for Don't Go There.

The first class-D amplifier based integrated circuit was released by Tripath in 1996, and it saw widespread use.

Class T was a registered trademark for a switching (class-D) audio amplifier, used for Tripath’s amplifier technologies (patent filed on Jun 20, 1996). Similar designs have now been widely adopted by different manufacturers.

The control signals in Class T amplifiers may be computed using digital signal processing or fully analog techniques. @fuzztone