Rendering can take so long that I'm not sure if rendering large numbers of animations on just your home machine is even feasable. This is why I got interested in render farm services once I heard they are a thing. And yet, at a glance it seems most render farms don't even support Houdini. Why is this? I thought Houdini was a popular piece of software. Why isn't it supported by the biggest render farmers?

Yes there are render farms that support Houdini: Grid Markets for one.

https://www.gridmarkets.com/ [www.gridmarkets.com]

They have a good video tut to help you get started.