An excellent quality commercial carpet or commercial carpet tiles would be good choices. Same with either genuine linoleum or glued down sheet vinyl. However, linoleum or sheet vinyl would mop up easily were there to be a spill from the washer, sink or the half bath. Genuine linoleum and most quality sheet vinyls are resilient and comfortable flooring options, but if you want commercial carpet or carpet tiles, then it's your house and you should get the floor you want.

Thanks for your comments.

I think my real question is if the built-up pedestal that the W/D will sit on is tiled with a drain under the washer, will that be enough protection in case the washer has a leak. If so, then I'd like to go with carpet for the warmth and comfort since I'll be using the room for more than just laundry. DH agrees with the builder /designer who thinks linoleum is the best option.

I assume cork would have the same limitations as carpet in the event there is a leak from the washer?

Natural cork tile or cork sandwhiched between two plastic layers glued down would clean up just like sheet vinyl. Your tiled W/D pedestal with drain would be suffice for floor protection from a water event, so carpet could realistically be considered. Just be sure that any carpet product is the tightest nap you can find, so stuff will sit on the surface and be easy to remove.

A couple of ideas to think about :
This guy did an amazing job of educating me on different types of flooring.
Hope it helps you make a decision too:

https://www.houzz.com/magazine/the-case-for-linoleum-and-vinyl-floors-stsetivw-vs~672851-Case-for-Linoleum-and-Vinyl-Floors

https://www.houzz.com/magazine/laminate-floors-get-the-look-of-wood-and-more-for-less-stsetivw-vs~621469-Floors--Get-the-Look-of-Wood--and-More--for-Less

I was in a flooring store and saw a product that caught my eye because of the color choices (grey and blues as well as the usual tile colors). It is a vinyl with a backing that requires no glue -- you just lay it down and it stays in place. She showed me the backing and it is something between the no slip cabinet liner and octopus tentacles with little round circles that grip and create mini suction cups (my description, not the manufacturers or sales persons ;-) ). I joked about how, in a flood, it could lift up, float around and go right back down -- but it honestly sounds like you could clean it up and do that.

I wasn't looking at who made it and have no idea how much it costs, but it sounded like the perfect kind of flooring for a utility room, basement, etc. I have to return a cork sample and can find out who makes it when I take it back if you want to check it out.