That's very interesting, it's similar but it doesn't sound exactly the same as the problem I'm seeing. These failed transfers don't cause any long-term problem for the connectivity, pings and such work immediately after the failed transfers. Furthermore, the transfers fail immediately, rather than after a large amount of data has already transferred. Even a small file such as omv-extras plugin can't even initiate the download on wget.
Well, mostly. The NIC appears to be just incompatible with Linux. Not a deal breaker, I'll leave my USB NIC plugged in, which is an easy enough workaround.
However, I got all four drives recognized!
For anyone else who is ever trying to get Linux, FreeBSD, or any other operating system that isn't Windows booting on their HP Mediasmart EX470 or EX475, here is the procedure:
Using a USB keyboard, after power on as soon as the health LED begins to blink blue and red hit DELETE and it'll become solid (either blue or red depending on what it was when you hit delete) and enter the BIOS. Advice on the Internet suggests to wait 5 seconds after this to ensure it's in the BIOS. Press F10 and then ENTER to save and reboot. Repeat this process until you're fully comfortable with the process of blindly booting into the BIOS and safely saving and exiting.
Boot into the BIOS again.
Press the DOWN arrow three (3) times to reach Integrated Peripherals
Press ENTER to select
Press the DOWN arrow one (1) time to select SIS OnChip PCI Device
Press ENTER to select
Press the DOWN arrow eight (8) times to select SIS Serial ATA Mode
Press PGDN one (1) time to select 4P(IDE)+4S(IDE).
Press F10 one (1) time to save and exit.
Press ENTER to confirm.
I figured this out by studying this video:
In that post, he lists out all the modes:
4P(IDE)+4S(RAID) => 4 Sata Ports in Raid Mode (Driver required, hotswap supported)
4P(IDE)+4S(IDE) => 4 Sata Ports in IDE Emulation Mode (No Driver required)
2P2S(IDE)+2S(RAID) => 2 'IDE' Sata Ports, 2 'RAID' Sata Ports *
2P2S(IDE)+2S(AHCI) => 2 'IDE' Sata Ports, 2 'AHCI' Sata Ports *
4P(IDE)+2S(RAID) => 2 'RAID' Sata Ports
4P(IDE)+2S(AHCI) => 2 'AHCI' Sata Ports
From other web searches, I saw that the top one, 4P(IDE)+4S(RAID), is the default.