Congratulations on your new place! Fun.
Insulated boxes for water pipes are great, but with a yurt, even once the pipes are inside they aren't safe from extreme temperatures. Adding heat tape will keep pipes from freezing, though then you are dependent on the grid. (AP states 1.5 million wo/power in New England after our most recent storm. Luckily temps are mostly still above freezing.)
If it were me I'd also work on a way to easily and quickly drain the system, if possible.
Personally I chose to just not have running water and love the peace of mind that brings, though a little primitive for many. And not sure I'd choose that with kids. Though maybe! Most in the world don't have running water - much less hot running water. It is harder to swim against the stream when everyone assumes there is only one way. Still convenience can be nice and for most, whatever effort it takes is worth it. But don't assume you need everything American society says you do to be happy. (While 1 in 5 are on antidepressants.) Test for yourself and see! (Wait! How did that soapbox get under me. Lol.
As far as enclosing under the yurt, I think that adds problems in addition to solutions. More mice and
. Warmer in the summer, perhaps? Definitely harder to access. For two winters I put bags of leaves around the perimeter and then a brown tarp. A lot of work, but looked nice. I would break it down in the spring. Last two years I left it open to see if it was noticeably colder, which I don't think it was. So I now just leave it open. Your platform doesn't look that high and snow coming from the roof may seal it up anyway. If those white things are the water pipes you'll want to protect them from snow sliding off the roof - even without the added cold protection.
Which reminds me of two other things. Anything stored under the yurt could become blocked in by frozen snow piles. So storing the kayaks and swimming noodles is fine, but not anything you might need - and definitely not the shovel you would need to chip away at that pile of ice.
Also snow sliding off the roof is VERY LOUD and initially terrifying. First few times you think the yurt is falling apart and your heart stops. Lol. But that is not truly dangerous. Always keep kids and pets away from the perimeter where the snow comes down if/when the roof has build up. Falling snow and ice chunks ARE very dangerous. Just something to be aware of. Overall - self clearing roofs in snow country are fantastic!!