Worth it, especially when the size of material is larger than 1''. Wet saw can prevent over heat to temper your listed materials depending on the thermal history. Very low carbon plain steel, say 1003/1005, may not be a concern though.

I see lots of hardness failures due to sampling. Regrind .05'' off surface with sufficient coolant often make them pass.

We had a long standing disagreement with a supplier of ours about hardness values on heat treated 4340 barstock. In the end, the root cause was that they were using a dry abrasive cut-off saw and not grinding enough material off, resulting in lower hardness values. When we would cut a section of the same bar using a wet abrasive saw and proper grinding practices, we'd get a higher hardness value.

I'd say that using a dry abrasive say will absolutely cause overheating of a material.

Short answer: No. But coolants/lubricants just help with the cutting process in general.

The surface hardness discrepancy you mention sounds very much like that caused by surface decarburization.

"Everyone is entitled to their own opinions, but they are not entitled to their own facts."

As per MagBen, the wet saw is worth it.