Yep, never underestimate the impulse of a mechanical connection discharging a capacitor. The risetime is very fast (subnanosecond) and the peak current can be, well probably just amperes in this case, on up to 100s or 1000s of A for electrolytics and, say, ones' hapless screwdriver (or breadboard, or..).
Note that pins in general will mate in some random order (well, maybe it's relatively consistent for a given pair of mates, but in general, you can't know which), so you're essentially guaranteed to get some full-magnitude unbalanced surge across one or both pairs.
Note that an unbalanced discharge does flow through the common mode chokes, so the risetime will be attenuated some; but within a few hundred ns, the core will saturate, and the bulk of the pulse will continue onward to the PHY. Which will in turn saturate in some ~µs, but in that time, the damage has been done to the PHY.
So yeah, ESD diodes are a must with PoE!
Tim