Of course that's true. But if you are talking about property damage only,
most people would rather collect their damages than their damages minus
one-third, forty percent, or whatever your or your lawyer's time is worth.

That may be a possible entity, but there is no legal requirement
to hit them up first.
A similar thing happens with auto accidents. An insured's collison pay
for his damages without regard to the insured's fault. Payments between
companies are governed by intercompany arbitration agreements which
assess fault between different insureds involved in a collision, and the
process is almost invisible to the insured. In the event that the
insured is determined not to be at fault, he is reimbursed his
deductible.

There's no requirement that you file a claim with your own policy first
in most states (barring no-fault states). I am perfectly allowed to
go pursue the at-fault party without involving my insurance company
here in Virginia. My insurance company even tells me this when I
file the report. Frankly, I always let them do it. It's what I
pay them for, and if they can't collect I'm still eligible to pursue
it (the only downside is if they settle for less than what I wanted,
but that's never heappened).