Is it possible (with reasonable effort) to use ethernet hardware to provide point-to-point digital I/O bridge?
Without thinking on it too much:

Send the data as UART-style encoding, with a start/stop bit (maybe from the UART itself)
encode/decode these UART-style byte-frames using Manchester coding
transformer-couple as normal for Ethernet over UTP

The issue with LVDS or RS-422 is that, by themselves, they don't define a DC-free code. So they don't work so well with transformers and other AC coupling techniques like those used with UTP.
That's why I suggest Manchester coding. It's what's used for 10Mbit Ethernet. It's fairly simple and yields a DC-free code. It’s feasible to use TTL or a small CPLD to implement it.
An improved version of Manchester uses differential coding. This avoids a number of issues, including idle all-0 / all-1 locking and polarity swap. Differential Manchester is also known as biphase-mark coding. It’s the protocol used for SPDIF / AES audio.
More here: What is the advantage of differential Manchester
It could also be possible to use an Ethernet PHY chip and make a logic bridge that converts signals to an appropriate MAC stream (MII, RMII), but I think the whole link partner establishment stuff begins to drag the 802.3 camel’s nose into the tent.