Replacing a Capacitor on an Onkyo HT-R960 A/V Receiver
The "tacky black goo" is normal. It is glue applied to reduce vibrations. The capacitors look fine. 

If the capacitors were bad, one of two things would happen:


An aggressive liquid would leak out, and cause the wires to corrode.  The copper wire would turn green.
The tops of the capacitors would pop.  Your capacitors have scored lines on top.  If the capacitors were bad, the electrolyte would boil and raise the pressure in the capacitor.  The can would burst along the scored lines at a relatively low pressure rather than exploding at some (much) higher pressure.


I think you need to look elsewhere for the cause of your problems.