Is `diagnostics` the recommended system for monitoring sensor data rate?

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I wanted to post as a comment, as I don't feel this is fully answering your question. But due to the size limitation I still posted as an answer.

I'll only briefly touch some of the questions, as they are pretty generic and thus would require quite a lengthy discussion (at least, some of them).

is diagnostics the recommended way to do what I'm trying to do

That ones easy, at least: Yes (in the ROS ecosystem, anyways) :-)

it feels somewhat verbose for a "simple" task

True, then again, you should not be using the raw diagnostics but use the diagnostics_updater as this highly simplifies the required code. It is not clear, if you have done this. There are also helper classes/functions that specifically deal with publishers (check the example, l. 191-211). Note that you also need to call tick() function on the respective updater for each publish. All in all, this should be like 5 lines of code, I don't feel that this is overly verbose.

for third party drivers, it would be convenient to be able to add topic rate monitoring without modifying the publisher

True (again). But there is no tool (that I know of) that can actually do that. You could achieve that by writing something like a generic "ExternalTopicRateMonitor" node, which subscribes to the topic. But this adds another node and one pub/sub-cycle, so it defeats the purpose of the diagnostics (at least, to some extent). It could still help you to achieve your goal.

is it possible to monitor nodes based on whether they have ever printed warnings/errors/fatal errors to the log?

I'm not quite sure what you expect here. In terms of "I'll just start up something", AFAIR, no. But then again there is the rqt_console, which does something pretty similar. So you could check how this is implemented therein. Or is that actually already what you want?

What about nodes that crash

Sure they cannot update their diagnostics. You could use a bond for that. But that would again force you to update the code. Or go the "ExternalTopicRateMonitor" way again (hoping that this doesn't crash) ;-)

I hope this answers some of your questions, but I'll also be interested in other people's opinion...