A manifold gives you access to *all your gas* when you shut down a post due to a failure like this:
With independents you cannot access that tank's gas. You're down to whatever left in the other tank. Fingers crossed its enough to get out.
There are risks in back mount as well, but there's no need to cross your fingers at all in side mount.
Assuming I have 260 cu ft of gas to start and 173 cu ft at turn pressure, and I need half that (86 cu ft) to exit, I haven't got a lot of time to identify the failure and turn off the correct valve.
A Mk 25 (or the Halcyon branded version) will flow 300 cubic feet per minute out of the end port - that's 5 cubic feet per second. That leaves me just 17 seconds to correctly identify and shut down the failed reg. Any longer, and I'm going to be dependent on my team mate(s) reserve to get out.
I have to take a positive action and accomplish it quickly to preserve sufficient gas to exit.
In contrast, in side mount starting with the same 260 cu ft split between two tanks and having the same failure at max p, I'd identify the failure immediately as it's either on my right or left, so the odds are I'll still close the valve in time to save some of the gas - although it's a moot point as I'd need to swap regs to use it, and I agree with you that it's a non starter of an idea.
However, on the way in (assuming I started with 3600 psi in both tanks) I will have breathed one tank down 1200 psi, then switched to the other, and breathed another 1200 from it, using 2400 psi total for penetration gas. But this also means I have 2400 psi in each tank at max penetration. So when the failure occurs and I lose access to all the gas in one tank, I still have exit gas in the other that is equal to my penetration gas -
even of I do absolutely nothing other than turn the dive.
Now, I'm also not a real believer in thirds in a two man team as it's not an adequate amount of gas in real world scenarios and was in fact designed for a 3 man team. But since more often than not I am in a two man team I plan the gas differently to ensure some extra margin for real world delays in turning the dive, depending on the dive, the conditions, tight passage, restrictions, etc.
Even on an easy dive, I still use a 1200 psi "third" with pressures over 3600 psi. Given that the average fill in cave country is now in the 3800-4000 psi neighborhood, this leaves an extra pad of 200 to 400 psi per side - about 13 cu ft extra with a pair of LP 85s at 3800 psi, and 26 cu ft with LP 85s at 4000 psi.
The same extra gas margin would apply in doubles as well, but 13 and 26 cu ft respectively only provides another 3 to 5 seconds to resolve the above mentioned problem.