The power supply schematic indicates this is a flyback converter, total output power rating = 80W (sum of output power rating of all rails), which is quite modest by today's standards. Your load, however, is only about 12.6W.
Failure of E7 (on the +5V rail) is quite understandable, as it was subjected to the most stress (ripple current). Even more understandable if this capacitor was several decades old - it would likely to have suffered "drying out" and its capacitance and ripple current withstand would have suffered as a result.
Failure of E13 (on the -5V rail) is less understandable, since there was no load on it, however, once E7 failed then E13 may have been subjected to higher ripple current and/or ripple voltage (and possibly DC voltage) than normal since E7 would have been providing the majority of the voltage clamping - details would depend on the magnetic coupling between each of the secondary winding pairs.
You mentioned that the replacement for E7 got to 50C - that is of some concern since it is not necessarily the temperature per se that does the damage, it is the temperature rise above local ambient. Check the datasheet of the capacitor used, which will provide guidance on the effect of (a) ambient temperature, and (b) temperature rise.
I would be tempted to replace E7 with two or possibly three capacitors in parallel, to get the ripple current stress down - ripple current is what casues most of the internal heating. Reducing ripple current by factor of 2 reduces the power loss by a factor of 4 (power = I^2 * R). Look up electrolytic capacitor datasheets to ensure the ripple current stays within manufacturer specifications. For these applications, the value of the capacitance is not as important as the ripple current withstand. There is a second filter after E7, E7 is intended to filter the flyback ripple current and get +5V supply voltage ripple to an acceptable level that is then cleaned up by L2 & C8.