Okay, it's beginning to look like your hunch that the origin of the reset pin oscillating is to be found at (or rather around) U553 may be correct after all.
At first I couldn't find it as you wrote it was the LT2925, but now I tried LM2925 instead and found the
datasheet.
I didn't know a voltage regulator with a reset-out function existed, but this is one. Since voltage regulators are normally very robust, I'd first start looking at the health of the components around U553, especially C556, C557 and C558.
Dried up caps may wreak havoc on the output of U553 and if that's not what it should be, that also generates a reset. It might also explain why it keeps resetting. Perhaps put a probe on the in- and output of U553 and see if anything strange is happening there...
Edit: I read through from the beginning again, and see that you already had a closer look at this reset pin.
What I might do to find out if U553 is the cause or if something else is the cause is desolder / cut the reset pin so that it's electrically isolated from the rest of the circuit and see if the behaviour has changed (on the pin of U553).
I might also desolder U553 and feed in 5 V from an external PSU and connect the reset trace to 5 V through a resistor and see if that changes anything.
What happens if you put another cap of 100 nF across C558?