As far as I can tell, you're doing everything right.
If this were sitting on my workbench, here's the next thing I would do:
On production units, I would make the resistor on MISO a pull-up, rather than a pull-down.
Page 37 of the datasheet only guarantees 100 uA on "MISO out high", but over 10 times that current on "MISO out low".
The 1.6 mA on "MISO out low" is enough to (dimly) light a high-efficiency LED with an appropriate pull-up resistor to +3.3 V.
I find adding LEDs to every questionable signal helps me find problems faster.
The 100 uA on "MISO out high" means don't expect the flash chip to work with a pull-down of 33 KOhm or less.
On my test jig (but not on production units), I would temporarily change resistor on MISO changed to weakly pull MISO to around 1.5 V -- that helps distinguish between high (near 3.3 V), low (near 0 V), and tristate (near 1.5 V).
I would re-run the test and make sure the only thing connected to MISO is the o'scope (or logic analyzer) probe and that bias resistor -- not even the PIC connected -- to rule out the possibility that the PIC is somehow accidentally driving MISO to GND.
I would make a custom test program on the PIC that does nothing but select the flash chip, attempt a READ IDENTIFICATION and reads 20 bytes, then deselects the flash chip, and then repeats forever. (It looks like maybe you've already done this).
It's theoretically possible that a PIC chip could be damaged just enough that it gives signals barely strong enough for the logic analyzer to distinguish "0" from "1", but not quite strong enough for the flash chip to distinguish them.
So I might check the voltages by (a) tweaking the custom test program so it runs the CLK at 1 Hz, so I can check each line on the flash chip with a voltmeter, or (b) running the test program at a more typical speed -- 500 KHz or 10 MHz should work fine -- and check each pin with an actual o'scope (not just a logic analyzer).
It's pretty easy to destroy a flash chip so it looks fine under visual inspection, but damaged such that now it won't ever work (always tri-state or always outputs 0).
Perhaps swap the flash chip with an "identical" M25P16 chip on something like the JeeLink and see if the problem follows the flash chip or stays with the PIC chip.
Perhaps re-build the entire circuit with fresh wires, a fresh PIC chip, and a fresh flash chip, and swap the chips around to see if the problem follows the PIC chip, follows the flash chip, or follows the prototype wires.
Best Answer
To simply read or write the contents in plain SPI mode without invoking additional functionality, tie both /WP and /HOLD high, most easily to the flash chip's supply line.