This seems intuitively that it would be true, but are there any stats on car age and accident frequency? Also where is the line on old? Vehicles made in the last 5-10 years will share most safety features, with all but a small minority of newer cars having some cutting edge. I'll admit I had been thinking of fleet age from the state of maintenance, but safety feature is a fascinating angle as well.. And the article you link in another comment was a fascinating read too, thanks!
Then there is the actual design of the vehicles, cars that were made in the last couple of years fare much better in collisions with cars that are older because they are a lot stronger, and are better able to channel the energy of an impact to areas away from the passengers. Many newer vehicles will for instance shift the engine under the passenger compartment in a frontal impact.
Older airbags had 'best before' dates associated with the igniters being somewhat unstable and that reduced the certainty of them going off as they got older.
Then, finally, the bodies themselves, once cars start to rust you can patch them up but they won't be the same afterwards. There is quite a bit of redundancy in the body but it definitely doesn't help if there is significant corrosion.
In fact I personally doubt (i.e. I have no evidence and am not actually making this argument because it would be "bad statistics") this is true if you compare Tesla's to the BMWs being sold into the same demographic. Those "safe new vehicles" you're positing include a whole lot of crossovers and minivans driven in much safer ways.