As you suspect, there are many parameters that can be tweaked and traded off in making any particular MOSFET. The ones intended for switching application strive for a low Rdson mostly. IC designers sometimes use "long tail" FETs to make something approximating a current source. I don't know what level of consistency and current flatness can be achieved by these devices (I am not a IC designer), but I do know they are used for this purpose in some cases.
No, there are no MOS current sources or anything else MOS in a 741 opamp. That is a strictly bipolar part. There are opamps that have FET front ends, or are exclusively made from FETs, but the ancient 741 is not one of them. Most of the modern low voltage "rail to rail" opamps are genarally all FET, like the Microchip MCPxxxx series.
I'm not totally clear about what you are asking, but yes, some MOSFETs in some cases are deliberately designed to exploit the roughly constant current behaviour in the saturation region. Like I said, I've heard IC designers use the term "long tail FET" for such devices when they wanted something that worked roughly like a current source. I haven't tried to search that, but you might find some additional information by doing that. I think the name comes from the fact that the channel is made long and thin, which causes it to saturate over a larger part of the operating range.
Whether there are discrete FETs you can buy that are optimized to exploit the constant current in saturation effect, I don't know. Probably any FET intended for analog (not switching) applications has at least part of the usable operating range in this mode. Switching FETs are optimized for low on resistance, so their constant current mode saturation region may be outside the normal operating range. For example, it might require more current than the device can handle or dissipate the resulting power. It's not that it's not there, just you can't reach it without desroying the device.