How can I turn on a vent fan over a gas range when gas flow is detected?
The switch linked in the question (Dwyer model V6EPB-B-S-1-B) won't work because its actuation requirements are too high. From its data sheet, actuation-deactuation flow rates for air in SCFM for the 1/2" pipe size valve are 6.50/5.00. (I'm presuming that they'll be similar for a non-air gas, ie fuel gas.)

An article at hunker.com asserts that "In most cases, all burners on the stove adjust down to about 1,000 BTUs" but also states "a low flame on the stove may only use 400 BTUs even though the burner might have an 18,000 BTU capacity." That's a little inconsistent but it's a place to start. There's no time unit specified but typically Btu is per hour.

The FAQs at eia.gov indicate that "In 2018, the U.S. annual average heat content of natural gas for the residential, commercial, industrial, and transportation sectors was about 1,036 Btu per cubic foot."

Combining these typical numbers, we estimate that a large gas burner of 18,000 Btu/hr will result in a flow of 18000/1036/60 = 0.29 cubic feet per minute. That's significantly below the V6EPB-B-S-1-B thresholds. The low-flow model V6EPB-B-S-LF actuates as low as 0.18, so it could detect the burner at high flow, but it wouldn't stand a chance if the burner were turned down to 400 Btu/hr.

Perhaps you could consider sensing something else such as heat or light from the flame. Or latch the fan on automatically when the gas igniter is energized, and use the more distant switch to turn the fan off when it is no longer wanted.