One reason is your statement "filing a claim negatively impacts your rates" is a blanket statement not applicable for all states. Some, like California, have statutes that specifically prohibit increasing premiums on customers that file a Not-At-Fault claim. Michigan, a no-fault state, requires that ALL collision claims on your vehicle go through first-party insurance (except for when a parked vehicle is hit I believe). YMMV.
Another reason (not premium related) against a third party claim - absent a judgment against the other driver, the third-party insurance is not beholden to you in any way. An issue is insurance being unable to contact the other driver and get their statement; until they do so, they will not make a determination on liability, not offer you a rental, not pay storage lot fees or tow fees, etc. Depending on how long that timeframe is, you could run afoul of your own insurance. If the other insurance ultimately denies liability completely because they never get a hold of the other driver (or its an excluded driver, there was non-permissive use, etc.), and you go to use your first party collision coverage, YOUR insurance may deny paying some of the storage lot fees, claiming you didn't reasonably attempt to mitigate the damages.
On the flip side, a negative to filing first party is limits on rental car reimbursements during repairs or total loss settlement. I'm commonly offered coverage for only 30 days max as a policy option, whereas Teslas take a while to get repaired (I waited 4 months in 2018).