The LM324 has a maximum offset voltage of 9mV (worst case, over temperature), according to the datasheet.
With your circuit, with 0V in, you could have a current of 9mV/500m\$\Omega\$/9mV = 18mA below which your pot would not be able to set the current. So it's not a very good design if you need to set it to less than 18mA. It's luck of the draw- the next op-amp (even in the same package) could be 9mV in the opposite polarity, so you'd just move the pot.
Maximum temperature drift of the LM324 is not specified (it's not intended for precision applications, after all), but it might easily be +/-10uV/°C, so if the board changes by (say) 70°C as the MOSFET gets hot, the current will change by 0.7mV or 1.4mA, so you'd have to readjust the pot. Of course the highest power dissipation occurs at high output currents, so the change is relatively small (1.4mA out of 2A is < 0.1%). A 20°C change in ambient temperature means a change of perhaps (no guarantees) of 0.4mA, which is several percent of a 15mA current. If you only care about 5%, and currents above 20mA, probably just okay.
Another difference between a cheap amplifier and a good one is the gain. The LM324 can be as bad as 25,000 gain (and it changes with temperature). A precision op-amp will have a gain in the millions. The difference will show up in how well it compensates for load or line changes (not a big deal in this case).
The bias current of the LM324 can be as bad as 0.5uA (typical 20nA) and it changes with temperature so if you had a high resistance pot, you could see it change with temperature.
The noise of the LM324 is a fairly miserable 35nV/sqrt(Hz), and it has nasty crossover distortion, neither of which affects you much in this case.
A couple of things (other than being extremely cheap) that the LM324 has that a typical precision op-amp may not have- wide supply range (especially on the high end), though it may not do so well at very low supply voltages, and it's single supply (input common mode range includes the minus supply) which you absolutely require for your circuit.
So there are plenty of reasons to use a decent op-amp if it's required by the specifications. Or you can get clever with the circuit- increase the sense resistor to get good accuracy for low currents, but to get wide dynamic range, a good amplifier (and other techniques such as good resistors and good layout) may be worth it. For just hacking around and if your current range not huge (minimum to maximum), an LM324 is certainly acceptable. There's no point in using a $5 op-amp if a 1-cent one will do. On the other hand, there are some requirement for which the best ones are not good enough and one has to resort to discretes and other techniques.
By the way, your circuit may not be stable against oscillation. It can be fixed with some passive components, but loading op-amps with the equivalent of a large capacitance in series with a small resistance is inviting trouble.
Your understanding of op-amps is fundamentally correct, and as Olin notes, an output with no load may very well drive close to the rails, but many parts will struggle even at no load.
What you may not understand is the models used for simulation, and these vary considerably in detail and accuracy.
This application note explains why most op-amp models are continuous time and why earlier models may not accurately show the limitations of the output. It also goes into some detail on how these models have evolved to bring greater accuracy to simulations.
Most interestingly, the model itself has no real relationship to the part in terms of the actual circuitry used, as the model is only representative of the behaviour of the part; these models rarely model start-up response (if ever) which can catch the unwary. (Chopper stabilised device outputs can be interesting for the first few milliseconds).
Understanding the limitations of simulation tools is critical in engineering, and only a thorough understanding of the parts being simulated (op-amps in this case) will save you from serious circuit mistakes.
The simulation is to help you understand the typical performance of a given part in a given configuration to help avoid many problems; you still need to understand the device fundamentals.
I have seen models (TVS devices in that case) that did not reflect reality and caused quite a lot of embarrassment when the box was subjected to lightning tests, because the designer had blindly believed the simulation.
Best Answer
It's a non-linear circuit that attempts to charge a capacitor when the input voltage exceeds the previously highest input voltage and, there is no basic problem here; it can't go unstable (as I suspect your engineer colleague thinks it will) because it's not a linear amplifier AND, the capacitor is only called into-action when a new high-peak of voltage arrives at the input. That occurs rarely and is of no consequence to stability.
There's a big difference between struggling to drive a capacitive load (which all op-amps suffer from to one extent or the other) and trying to remain stable in the presence of a capacitive load in a linear circuit. This rectifier circuit WILL NOT go unstable unless you have chosen an op-amp that cannot be used in linear unity gain situations (there are a few).
You do need a "light" pull-down resistor in parallel with the capacitor by the way.