The powersupply swap could have accelerated it's demise. The old power supply might have had sagging voltage levels and once replaced and voltages brought back to spec, dying caps probably roasted at a much faster rate. Hard to tell since you didn't have that SE for very long. It could have just been a coincidence who knows.

But truth be told, psu swap or not, sounds like your Mac would have died like this either way. Yeah recapping it "should" fix it. If not, then solder reflow might also be needed. I've heard that could also be an issue. But I've heard not as big an issue for the SE as it was with older fanless Macs and the newer Classic/Classic II line which also are known for failing analog boards.

Worst case scenerio is a transistor or some other part on the analog board died. Sometimes running a mac with bad caps can have some bad consequences. Hopefully all yours needs is new caps.