I’ll respond to each point in line.
Our board does get to 60C measured with a thermocouple, so that could explain some of the issues we’re having.
GigaBlox Nano getting to 60°C isn’t the issue, it’s expected as that board doesn’t ship with a heatsink. Some of our customers put a 10mm x 10mm heatsink, but generally the board runs fine without it.
Gigabit nano 1 connects devices 1-3 and then is daisy chained to the other gigabit nano2.
There is a known bug on GigaBlox (that also applies to GigaBlox Nano, as they use the same chip), related to daisy-chaining, see below.

Are you using the same numbered port on both for the daisy chain connections?
I’ve read that only voltage mode line drivers work with this switch, but I’m having trouble understanding what the LAN7500 is. To my understanding it looks like the LAN7500 could be current mode.
Yes, LAN7500 is current mode. You can tell this by looking at the dev board schematic for LAN7500, see how the transfer center taps are pulled up to a 2.5V rail. That means the LAN7500 PHY works by pulling current from the 2.5V rail into the transformer to generate a voltage on the other side. This is in contrast to voltage mode, where the PHY itself generates the voltage directly and sources/sinks current in and out of the transformer.
But if it is current mode, why are we able to use the switches at all?
What you have right now is the LAN7500 directly connected to GigaBlox Nano without a transformer (please correct me if I am wrong). You always need a transformer when using a current mode PHY, so I would not expect this to work. The fact that it does work intermittently is not super surprising either. Ethernet doesn’t always fail in a binary way, and the signals are probably still being received/transmitted on both sides, but I wouldn’t expect it to work reliably.
In summary, you need transformers on your current mode PHY, otherwise I wouldn’t expect it to work reliably.