I switched to Arduino Uno. Wiznet's library does work with the Seeed W5500 shield when using Uno. The speed is 139.99 kbytes/sec. For comparison, Ethernet 2.0.0 gets 329.00 kbytes/sec on the same test with Uno, and 689.69 kbytes/sec with Due.
I also retested W5100 (Arduino Ethernet R3). Wiznet's library get 10.17 kbytes/sec (yes, slightly faster than 9.83 kbytes/sec it gets with Arduino Due). Ethernet 2.0.0 gets 82.66 kbytes/sec when using W5100 on Uno, and 109.73 kbytes/sec with Due.
Without a doubt, Ethernet 2.0.0 is much faster than Wiznet's library.