With most electronic loads it is not about getting the full specified Ids. That number is more like a theoretical thing anyway, calculated form R_on for the cold FET and the theroretical cooling capacity with the case held at a fixed 25 C. So a cooling level that may be reached with water cooling at best.
The upper limit for the get voltage is purely a destruction limit. There is usually not much gained from increasing the gate voltage to higher than some 10 V even if many fets allow 20 V. This only gives a marginal reduction in R_on.
There is usually curve for Id vs gate voltage. E.g. a typical IRF540N (infineon DS) would need some 4.5 V to reach 10 A and some 5.1 V to reach 30 A which is close to the specified limit (33A).
The Id vs gate voltage curve is not linear, but more gradual at the start (more like (Vgs-Tth)² ) and than get again shallower at high currents, as part of R_on *I is subtraced from the effective gate voltage.
It depends on the use of the electronic load. For low voltages, like a single NiCd cell one may indeed be limited by R_on and want thus high current, low voltage types.
For higher voltages, like > 5 V and maybe already with a single Li cell one is more limited by the heat. Here in addition to the DS Ptot limit there is a more practical limit for the cases, e.g. some 30-40 W for a TO220. So with 4 V this would be some 8-10 A max. for the TO220.
Especially beyond some 20 V the SOA can be limiting and R_on or the maximim Id is not that relevant. Especiall lower voltage MOSFETs are usually optimized for hard switching and can not withstand as much power if the voltage is high (e.g. >20 V). There is a chance that they fail due to internal thermal instabiltiy similar to the 2nd breakedown in BJTs. A problem here is that there are quite some false SOA curves out there, especially those that show just a power limit and not an extra reduction at some voltage, like 1/2 the rated voltage. The get a reasonable good real SOA one usually goes for higher voltage (e.g. 200-600 V) types even of the actual votage used is only some 20-40 V. There are special type tested for linear operation, but these are quite expensive and usually also higher voltage types.