I used to work for the USFS as a Wildland Fire Initial Attack Module Leader in SE Utah; Moab/Monticello Ranger District of the Manti-LaSal NF. In my years there, I never heard of any regulation requiring folks to carry fire suppression tools with them nor any LEO citing people for not having them onboard. That was for the entire Moab Interagency Fire Center; USFS, BLM, NPS, and Utah State. We did stumble upon a few old 60's-70's vintage posters in the shop recommending a shovel, bucket, and axe to visitors though. Never hung them up on the forest and they were just kinda nostalgic.
I'd strongly advise not carrying any tool, especially one as dangerous as an axe, unless you are proficient with it. Improper use can be a serious safety hazard and I've worked several incidents where the novice with an axe stuck it in his/her leg or foot. Wicked nasty wounds!! Remember the old saying- just owning a hammer doesn't make you a carpenter...