I endorse @harding’s math and that does sound like an elegant solution (also a great write-up advocating for it!). I had not thought of the idea of including the ping frequency with each TagTime log entry. Super smart.
Here are my remaining reservations:
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As @charles99 predicted I’d say, the universal ping schedule is really nice. Like how literally 7 or more devices in our house all ping in unison. So handy! Most users won’t care about that but it’s a nice feature for people who do eventually get that hardcore about it and it’s a shame to break that.
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This one sounds ridiculous but I’m constantly amazed how true it is: Giving users choices is usually bad. I’m not sure how to articulate this right now if you don’t already know about the UI principle often pithified as “don’t make me think”. Or chapter 3 of Spolsky’s treatise on UI design. Ping frequency may be a very reasonable thing for users to choose but you’d have to be sure they’re equipped to make tradeoff between accuracy and annoyingness of getting pinged in quick succession often.
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This one might be bogus but you can build intuition about ping frequency if you stick to 45 minutes. Maybe bogus because, as @byorgey was saying, your intuitions may persist in being wrong anyway.
Finally, a question for @harding: I agree that computing the expectation of the true amount of time spent is straightforward (just add up the frequencies for each ping) but does it get much messier to compute confidence intervals on those estimates?