Batteries do not discharge slower than a capacitor - like capacitors, they supply the demand. (That's if in parallel. See below re individual discharge rates.)
Similarly, capacitors do not charge faster than a battery (assuming burp and shortish overload situations) because BOTH are at the same voltage (as you stated).
For long burps and overloads, it is pointless having a capacitor - they have minimal reserve time (even a 12V 1.2AH battery outlasts a multi-Farad cap).
If the voltage drops from 14.4V to 13.0V for 5 seconds due to an overload (ie, the alternator is of insufficient capacity OR due to cable drops), there is no recharge involved though - ignoring surface charge - the battery is still being charged (since 13.0V is above its nominal fully charge voltage of ~12.7V).
REPEAT - there is no recharging. Both the cap & batt have dropped to 13.0V.
Further to that, the battery's surface charge will be far greater than the cap's capacity. The advice is usually to use headlights for 5-15 minutes to remove surface charge from a battery, or let it rest for 24 hours. That drops its voltage to its "real" fully charged voltage of ~12.7V (in theory, 12.67V). Will a cap last 5-15 minutes with headlights and only drop to 12.7V?
Unless you are going into certain material or chemical issues (which are not relevant to the situations discussed), your comments -cum- assertions are simply invalid - eg:
The cap voltage will be the same voltage as the battery. (I am assuming the cap is parallel to the secondary battery(s) that are next to the amp. The alternator voltage will be higher.)
Charge and recharge IS current flow.
A cap is not an open circuit - it is a cap. (A discharged cap is a instantaneous short circuit when voltage is first applied - hence the need for series resistance to limit the current if the short or high current could damage the supply.)
And a cap is not an open circuit (through it) for AC voltage. IE - it acts like a filter making the voltage smoother.
But the battery also exhibits cap-like behavior. (Some circuits even model the battery as a very large capacitance.)
There is no debate per se in the above.
The inferences drawn are simply incorrect. How can a cap recharge and discharge from the same voltage? It recharges when its rail voltage is higher than itself. It discharges when the rail is lower (ie, the water and bucket model). That's no different than a battery.
As to discharge rates, I don't know where that comes from.
The only truth to that is that a capacitor discharges (ie - its voltage decays) much faster than a battery because it has nowhere near the same capacity - ie, capacitance. (That does depend on sizes, but if a 9V battery keeps a lamp lit longer than a 1F cap...)
Ergo, why use a cap?
You may see several articles that show how a small battery (eg, 12V 1.2AH) has far greater capacity than a 1F or larger cap.
Hence the argument with a battery and cap in parallel becomes - for greater capacity - a battery with a small battery in parallel.
That then becomes 2 equal batteries - or a larger battery - for even more capacity.
The main desire for capacitors would be to replace batteries as caps do not have inefficiency (well, maybe 1% compared to ~30% recharge inefficiency for a battery).
But to replace an automotive battery would require a trailer load of capacitors.
Apart from the cost involved, it is simply not worthwhile. Cars burn far less fuel for the battery's 30% inefficiency than towing a heavy trailer around.
Another reason for caps might be because secondary batteries are undesirable or can't be located near the amps, but that is not relevant in your case.
Unfortunately I have lost the link to the original article, but have a look as Richard Clark's article as reproduced at
http://forum.realmofexcursion.com/accessories-electrical/17919-why-you-dont-need-capacitor.html
FYI - Ignore any mention elsewhere re "placing more load" on the alternator. The reality is quite the opposite - bigger batteries (and caps) LESSEN the load on the alternator - not that an alternator should care, though undersized batteries else alternators can cause noticeable lifespan issues.
Also, despite people saying otherwise, ripple on a DC supply is AC. In this case, ripple etc is an AC voltage offset by 12VDC (or "AC with a 12VDC bias").