Well done fixing it !
One trick you could do to have found the defective/shorted capacitor, without trial unsoldering, would be to have left it in the state of overload with the regulator supplying 1v. (Or power the board from a current source or low voltage with current limiting) Then with the meter, look for low mV voltage readings on the board. For example with the meter placed on the terminals of that shorted capacitor, it would likely have had a slightly lower mV reading than the other capacitors on the board, because of the higher currents in the tracks and the Ohmic resistance of the tracks, leading to the shorted capacitor's terminals.
I have heard of people applying the correct board voltage from a high current capable source, then the defective tantalum lights up like an Xmas tree letting out the smoke, but I have never subscribed to that in case it vaporizes the pcb tracks.
( I can tell a funny story about Tant Caps. Many years ago I was doing a research project on neonatal breathing and had set up some breathing monitors in a neonatal unit of a local hospital. I had built the monitoring equipment. But we needed a way to record the analog breathing signals. So the Professor in charge of the project offered to loan me his Tanberg reel to reel slow speed tape recorder, I agreed, so we set it up in a nursery monitoring some babies, and we went home for the night. The next day when I arrived for work I was told that my breathing monitoring equipment had "caught fire" and they had to evacuate the unit and the fire Dept came. Gobsmacked I went to inspect the equipment, it all looked ok, except for a "smell". It had turned out that one of the Tantalum capacitors inside the Prof's Tanberg tape deck had smoked, and the smell of it was so bad that everyone panicked and assumed fire. I tried to explain it wasn't my equipment at all that had troubles, it was the Prof's tape deck but barely anybody would believe me).
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