So you could implement a chip which reacts like an official passport. When the border guards see that the signature is invalid, you can explain how it's just a prank and you'll all have a jolly good laugh about it.
There are also optional subprotocols that allow the chip to be authenticated (i.e. proof it knows a private key). These prevent copying valid signed data to a different chip.
NFC chips can be locked. That means the data can't be overwritten. No matter the writer, nor its strength, you can't overwrite a passport's chip.
I suppose you could use an EMP - but that would ruin a lot more than just some trips.
Both sides even have the info printed. One side in human format, the owner side in machine readable.
For a random traveller you can probably guess roughly how old they are, which is a few bits for the date-of-birth, and maybe you could strike up conversation and discover their name (or maybe it's printed on baggage, called out by fellow travellers etc.) but yeah it'll be very hard
For a very well known person you can likely discover everything except the passport number and you might get a decent guess at that from knowing roughly when it would be issued.
From a very well known person you could probably also steal everything you need directly, if your purpose is to create damage.
I'd consider that pretty unlikely. Degraded chips would most likely provide no signature, not an invalid one. (Being able to randomly flip bits would be a big security problem for these kinds of ICs, so I'd assume they'd have robust protections against that.)
I don't think ICAO passports can ever be rewritten post-issuance. Some national IDs can, e.g. to change the holder's residential address, but for passports, I don't think any part of the on-chip data can be changed post-issuance, since it would also require re-printing data on the photo page usually under protective plastic.