SVS gear is generally engineered fairly well, so the odds of the speaker being damaged are small. You can expect years of use with a new plate amp if the speaker is fine.
Most gear these days use switching power supplies and all it takes is one bad capacitor for the whole thing to stop working. If the LED comes on, then the power supply is likely working, which is worrisome. Caps don't usually get burned or black. They tend to bulge out or leak when they fail. The burn marks are likely from a component next to it but I am just guessing without seeing it first hand. If it's just the cap, it's an easy repair. If something next to the cap went up in flames then you're looking at a lot more work.
If it's just the cap, caps are cheap and there are plenty of tutorials on desoldering and soldering on YouTube. If you're new to repair, you will need a soldering iron, solder (electronic, not plumbing) and a desoldering tool like a manual suction tool or solder wick (fine braided copper wire). The value of the cap will be on the case, so wipe of the charring with alcohol and get a cap of the same value and size. If you can't read the value, get it from the new amp.
If you don't mind investing in the parts and equipment, this can be a learning experience. If replacing the cap doesn't work, it's no big loss since you have a new plate amp. Do some homework first, though. Capacitors hold a charge so even if the amp is unplugged there can be high voltages present. Any large capacitors on the power supply or amplifier stage need to be discharged before you can safely work on circuitry. You also need to connect it to the mains (120V) for testing so proper precaution needs to be taken. If you are not comfortable working with potentially dangerous voltages, then leave it to the pros.