underfloor heating and solid wood floor

I presume that "Wet System" means water filled tubing or piping. But oh yeah, wood is no problem & is extremely common, both nailed-down traditional floors & floating laminate floors. There's no steam or moisture being released & the floors don't heat to anything more than a hot summer day.

However, insulation is okay but you really just need a shiny space blanket type of radiant barrier (the shiny side of tin foil) to reflect most of the heat upward & not even consider nor attempt thermal loading. You'll be happier & more comfortable with a system that can heat up & cool down or just maintain a room's or building's temperature.

I have used hydronic heating on several wood floors and they have lasted the same as over tile or embedded in concrete. I would insulate under the pipes because heat rises but you will be heating the crawl space more without insulation. The one thing I learned on my first house I did this was you don't want to have a big set back because it can take several hours for the heat to stabilize I found a setback of more than 10 degrees caused the flooring to squeak more even after the house had stabilized. If I left the temp constant at 72 degrees after several days there was no more squeeking. The first home I did this on had ceiling heat never made sense to me but electricity used to be dirt cheap in pet Pacific Northwest, I changed to a gas boiler and under floor and even having my heat at 72 100% of the time my total power & gas bill was cut by almost 70% and it felt warmer in the house with warm floors.