I ripped the demo, rebuilt as an .ISO, tested in PCSX2 and it is fully functional, and in all it's original glory with early textures, logos, etc.
The BETA itself includes Practice and College, and the main menu is fully scrollable with nothing blocked out, including Network Play. The total size = ~213MB.
Video of gameplay and options of BETA, using PCSX2 1.4.0: https://www.youtube.com/watch?v=Y4Zw-MsbZLU
Video of gameplay on actual PS2 (loads DEMO): https://www.youtube.com/watch?v=mder8EaYMNc

DOWNLOAD:
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Mega:
https://mega.co.nz/#!1cFQnCgY!7PNqdg0t52bHiTwEooDGJOPe0Lf3eCsyNWA0uCchINE
Mirror:
https://mega.co.nz/#!0FEizTgS!1SfhMqUXe7pi4x_tfA8QPpHy00xxysvYNLTOtXP5sD0
TPB (Torrent):
https://thepiratebay.se/torrent/11801366
Magnet (Torrent):
magnet:?xt=urn:btih:273aa8874183090dd8da9ad9d5c2d2ec86268d9c&dn=Tony+Hawk%27s+Pro+Skater+4+DEMO+PS2+Rare&tr=udp%3A%2F%2Fopen.demonii.com%3A1337&tr=udp%3A%2F%2Ftracker.coppersurfer.tk%3A6969&tr=udp%3A%2F%2Ftracker.leechers-paradise.org%3A6969&tr=udp%3A%2F%2Fexodus.desync.com%3A6969