There are no hard and fast rules for which pedal type you want to use on what bike. I am of the opinion that riding a bike is a personal experience and how ever I want to configure the bike for my purpose is my choice. Others opinions are just that.
My MTB/commuter bike has those pedals and I like them a lot. My MTB shoes are used with that bike. All my road bikes, some of which I use for commuting, have Look Keo pedals using road shoes.
If I plan on doing something in addition to going for a ride that involves walking, I take the MTB/Commuter. I don't have a gavel bike yet, but it would likely use the EH500 too.
Note that there are multiple SPD pedals. Shimano designates the road version as SPD-SL.
https://www.cyclingweekly.com/fitnes...s-pedals-20941