The idea of LC filter is to clean up high frequency noise from other components or switching regulators. This rail is then used as power for the ADC/DAC and a voltage reference (as is the case on my small pin count MCUs)
You can use that filtered rail to also power other analog parts, assuming it wont inject interference back into the rail of course. A steady potentiometer or stable opamps should be completely fine to add. Alternatively if BOM cost is not a big concern, you could also duplicate the LC filter circuit and that for your analog parts.
Whether its necessary is hard to say. I wouldn't expect it to influence 10-bit performance all that much. But it also depends how noisy the original power rail was and how the analog circuit was implemented.
For example a potentiometer circuit has very little "Power Supply Rejection Ratio" (PSRR). If you then buffer, amplify, etc. that signal, even with opamps that typically have very good PSRR values, it will still remain noisy signal.
You would have to clamp the bandwidth (as with almost all analog signals) to make it less noisy. That could be as simple as adding a small capacitor between the wiper terminal and ground (or whatever reference the potentiometer uses) to throw out all that HF junk, or add some RC low-pass etc.
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