Electronic – Do it yourself operational transconductance amp

Using a resistive differential divider as the input to a differential amplifier with a lower voltage rail than the Vin being measured is valid BUT be aware (and some simple calculations will show you) that the common mode voltage does not cooperate properly with the division and you get increased error.

AFAIR error effects due to errors in the divider resistors act on the full common mode voltage whereas the dividers themselves divide only the differential signal. So any errors in the divide resistors effectively increases by the error % of the resistors. If your common mode voltage is large compared to th edifferential signal "there will be problems".

Very approximately if you have 20:1 divide and 1% resistors you can get ~= 20% errors. Using 0.1% resistors as @The Photon suggests may seem a luxury but isn't and will probably only deliver accuracy about as good as you have said is OK if unadjusted. Note that this error may be removed by either physical calibration or software adjustment and you are then just left with the time variant effects. eg a 25 ppm/K resistor may end up looking like 500 ppm/k or perhaps 1% error at 20 K variation.

All the ab0ve is approximate as you have to see how combining worst case pars works in the circuit you use, and eg temperature effects will usually track and so partially cancel to a large extent.

If you can provide a floating high side supply you can buy isolation amps that have specs better than you need.

Here are Digikeys reasonably wide range of offerings

You can use these elegantly clever dual linear optocouplers
to form an isolated linear interface.
This is Vishay IL300 but there are others.
The circuit relies on the balance in the two receive diodes- ne being used on the send side for feedback linearisation and its (hopefully) good-twin on the receive side for output.

In simplest form

Usually done more like this, but still simple enough

Why are Op Amps more noisy than other types of amplifiers (BJT and MOSFET, for example)?

I doubt this is true, especially if you consider the noise factor for the same circuit gain (but another answer might give more concrete information).

So, in short, why do OP Amps have electrical noise issues (if at all), and what is the source?

TI has a couple of good app notes on this:

Chapter 10 of Op-amps for Everyone

Noise Analysis for High-speed Op-amps

In short, the feedback resistors and the op-amp's input stage transistors both contribute to noise.

Perhaps Op Amps lack sufficient filtering capabilities that are inherent to BJT and MOSFET design?

Op-amp circuits can be filtered to whatever bandwidth you like. It only requires adding an extra capacitor or two to the op-amp circuit, or for higher-order filtering adding an additional filter stage after the high-gain stage. Which would be equally required for a discrete circuit.