Update -- I managed to fix it in the end!

I swapped out the 10V 2200uF for 16V 2200uF (all 4 of them in one go) and this solved the problem. Although none of them appeared bulging, various things I've read on the internet seemed to indicate that this was the sort most likely to fail due to the low voltage tolerance range, and it certainly seems to be true in this case.

Opening the box and removing the power supply was easy; desoldering and re-soldering the capacitors was a little more tricky but no real difficulty as I've done soldering before. The big win was ending up with $200-worth of TiVo operational again, so a 3rd TiVo for our house to go in the kids' playroom.

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The power supply board removed
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One capacitor desoldered. Make sure that the -ve of the new capacitor goes in the correct hole (on the shaded side of the circle)!