No, I have not suggested that 14 bits is enough. I can't suggested any bit count at all. You have not supplied any information about about what you are measuring, and what your requirements are, so no reader on this forum can suggest how many bits of resolution you will need, or what sampling rate you will need.

All I said about 14-bit ADC data is that up to 14 bits are easy to send in binary form since you have two bits left in two bytes and can use a zero in one byte and a one in the other byte to handle the framing, letting the receiver know which of the two bytes that transfers the low and high bits of the ADC data.

That you are going to perform mathematical operations on the signal, efter having read it does not automatically imply a need for a different resolution. Next issue is if the microcontroller or the PC will perform the math operations.

Having a regulated power supply doesn't help. It is a minimum requirement, but not enough. A power supply produces a lot of noise. Even if you use a high-end power supply, the current spikes from the varying load of the processor is enough to produce noise because of the resistance in the PCB traces. That means that you need to filter the VCC. And you need to filter any voltage reference.

Another issue is that you must separate resolution from precision. Having a 10-bit or 12-bit or 16-bit ADC just changes the resolution. Noise on VCC or voltage reference will generate noise in the samples and can make one or more of the bits meaningles. The precision of the voltage reference will affect the scaling factor of the ADC - with a 2% voltage reference, the ADC full-scale readings of a 10-bit ADC can be 20 ticks wrong. With a 12-bit ADC, the full-scale reading can be 80 ticks wrong. With a 16-bit ADC, the full-scale reading can be 1300 ticks off. Then you have to add the noise figures. Having 2% noise on the voltage reference will add just as much error. The ADC may also have a couple of bits of non-linear and quantization errors.

Without knowing more about your application, we don't know what needs you have when it comes to precision. And to resolution. For comparing trends, it is normally enough with a lot of resolution, as long as the resolution is repeatable. For absolute measurements, you also needs good precision, where you have control of the scaling factor, offset and linearity.