What you're talking about sounds similar to something in Spanish law (to my understanding) called "servidumbre de paso".
Basically, if the property is completely surrounded by someone else's (or multiple people) and there is no way for them to access public space from it without stepping on the other's property, they may request a "servidumbre", and unless of previous accordance, the one that needs the way will pay the other/s for the right to traverse their property, in the least damaging way, or the shortest, etc, with some criteria to consider before it being granted.
So with your first example, if this worked similarly, the government would request something like this, and pay the owner that is surrounding them for the right to pass, but it would have to be in a very restricted manner (like just to get in and out, without damaging anything), but I guess it still would not be open to the public, just to maybe get in there and access it for maintenance.
In regards to private ownership of a body of water, I guess the question for me would be on its mass or volume, or surface it takes, i.e: having some criteria, that if surpasses x metric, cannot longer be owned privately.
But, if owned by the government (and so of significant mass/etc), there should be minimums of maintenance they should do, and if it impacted negatively the surroundings, they should pay for it.
All of this would require very specific law and metrics to follow, but the general gist. With that said, I don't like a private owning a large mass of water (like a lake), and there's even regulation on subterranean water as well, but we're not getting into that.