Is this the advanced key (i.e. you can keep your key in your pocket)?
If so, holding the key up to the key spot energizes the NXP chip inside the fob so that it can handshake with BCM2. So, if it doesn't work while the key is not near the spot, you either have a problem with the RF circuit inside the key fob, or power to that chip... The same battery powers both the remote keyless entry (button press) as well as the NXP chip. If the button press works but prox start does not, then I'd take a look at the circuit board. You can pull the board out after you remove the battery carrier. It can be a little tricky, there is a detent in there but by grabbing the edge with stout toothed tweezers and wiggling it, it should come out just fine. The remote keyless entry (button press circuit) is on the opposite side of the battery, and the advanced key/prox circuit is on the battery side. \
I'd clean the battery contacts with a cotton swab and iso alcohol while you are in there, and if the battery contacts seem loose (test fit a battery with the board out) you can bend the base contacts up slightly (very carefully! not much tension) and bend the outer contacts inward slightly with your fingers.
Take a look at the coil in the prox circuit- It is a large square inductor looking thing, usually a white top, with winding visible on the sides. Gently grab it with your fingers and try to tilt it in all directions - Has it come desoldered from the board? Inspect the solder joints for all these components on the battery side - Is there any contamination visible? Do any of the solder joints look cracked?
The components you will see on this side: The (probably white) square coil - Prox coil; Small brownish 2-terminal device - Oscillator for the prox transmitter circuit; Small 8-pin chip - Prox transceiver IC; Small 3-terminal device - Prox RF front end transistor; Fat, rectangular chip with a bunch of pins on 2 sides - This is the NXP PCF7945AC chip with integrated antenna. You will also see a number of small (0402) components, and 3 of them are on the antenna trace at the end of the circuit board - You will see an interesting looping design of circuit traces on the opposite side of the board from the battery. Are all these tiny devices intact? There will be spaces for around 8 of them, but only 3 are populated, you can tell if one has broken off the board because it will leave indentations in the solder on the tiny pads.
Anyway, maybe visual inspection will reveal nothing but if holding the key up to the key spot still starts the car then this indicates that your NXP chip is still functional and maybe something is wrong with the prox transceiver circuit. That black rectangular chip with all the pins on 2 sides is actually capable of starting your car on its own - i.e. you could cut it off the board and hold it up to the key spot and it would still start the car. So if that's working, it seems that the problem must be outside of that chip.
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