Help understanding high \$V_F\$ ratings of Schottky diodes
If  you run a diode near its rated current, you don't get much advantage, but that has little to do with the process involved. Instead, the limit in such cases is the power dissipated by the package, and for the same current level that implies the forward voltage will be about the same. So the Schottky diode will be wimpier (for the same current) and will not perform much better.

Instead, look at what happens if you run at, say, 10% of rated current. Compare this 40V, 1A Schottky with a classic 50V, 1A 1N4001 but with both run at 100 mA. The Schottky will drop about 0.4 volts, the 1N4001 about 0.8 volts.