The input of the amp alone should read probably 10K or more? You are normally just measuring across the volume control pot.
If the bluetooth module is connected, then the output of that may show lower resistance, especially if it is powered up and not muted - the output will be actively trying to hold the correct signal level to drive whatever it is connected to.
It's not actually an "ohms" reading if there are semiconductors or especially active electronics involved, as the external circuit can clamp the voltage at a point, or force the voltage on the probes to be similar, making the meter "think" it is connected across a low resistance.
[To read Ohms, the meter puts a small current through the probes and measures the voltage that produces across whatever is connected, in effect doing an Ohms Law calculation to determine the resistance that would cause the voltage drop].
Technically, a direct connection rather than bluetooth should give rather higher quality, at that point at least. Bluetooth stereo audio is quite heavily compressed using a lossy system similar to MP3.
The loss of quality is even less than with 320K MP3, so it is a rather minimal effect, not generally objectionable when using such as bluetooth headphones.
Whether it's possible to hear any difference using a direct connection instead depends on both the quality of the audio source [never compressed using a lossy system such as MP3 etc] and the amplifier / speaker noise levels and frequency response; very unlikely, with a simple amp and open speaker as you show above.