W5500 with POE functionality Link status ON with nothing plugged in?
If you based your design off of WIZnet's &quot;wizarduino&quot; schematic, then I suspect you didn't notice the note on it that said not to actually populate the 3 pull down resistors On the PMODE pins.  These are resistors R30, R31, and R32 on your schematic.  The device will not work correctly until those 3 resistors are removed.  With the pull down resistors populated, those 3 connections are just voltage dividers, ones that will ensure an undefined state (neither high nor low).
Considering they control auto-negotiation behavior, it does not seem like that big of a leap to think that undefined states on these pins would cause the chip to lock up in an unresponsive state with a link active LED on due to the logic getting screwed up.  It would be about 1s after boot when it would begin trying to detect if there was a link or not.
This isn't necessarily the only issue, but any other problem discovered would not fix the problem until this undefined pin state problem is also fixed.  Let me know in the comments if the problem persists after removing those 3 resistors.  At least its just a quick desoldering job, right?  We should all be so lucky to have those kinds of problems!