The 'free floated' on an H&R is different than on, say, a bolt action. On the bolt gun the action is well bedded or glass bedded into the stock with perhaps only the first couple inches of barrel forward of the action bedded, thereon it has no contact. In fact, for practical purposes you could cut the stock off right there but we must hang onto something out there. On this design where we hold it forward has some, but little effect for practical accuracy.
On the H&R you have that forestock mounting stud. Yes, from the spacer forward you can have no wood touching the barrel but you cant avoid that stud mount and whatever/however it affects things. The real pressure issue here might depend upon the pressure between the stud and the spacer face to the action more than the screw tension holding the forestock. I sure cant argue with the results some are getting by fine tuning that stud screw, but most are also using well fitting forestocks dedeicated to each barrel. They undoubtedly work in conjunction, but one should have more affect than the other, and Im curious which it is.
I believe this requires more well conducted trials, but if the action/locked up barrel has no pressure from the spacer then the stud thus has no forward pressure (as would be the case from a tight spacer to action) then the barrel harmonics should be better and as unaffected as possible on this design. I know I like how mine shoot when the forestock is not a tight fit. The recently posted spring loaded plunger from the spacer to action may be the next step from my basic approach.
But then we have to hang onto something out there and changes the above again. With no other tensions induced on the assembly than the hand support from straight beneath any affect may be minimized. I know for myself that since I shoot a lot offhand that I am a far greater cause for error than anything, so I can forgive a bit on the rifle. My forward handhold is not way forward like some, about my farthest is with the forestock screw just ahead of my index finger, and usually well back of that (comfortable and natural for me).
So, in terms of priortiy I think of it like this:
Snapping the action closed with the same amount of force each time
Have a clean and dry latch and shelf for lockup
A no tension forestock
Get comfortable and make a good shot each time