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Numerous online postings perpetuate the myth that our cars' auxiliary batteries supply the cranking power necessary to activate the engine's ECO stop-start function. I tried fixing the ECO stop-start function not working by following this readily available advice and ended up wasting nearly £100 by replacing a perfectly good auxiliary battery. When I purchased the vehicle, the main AGM battery was in pristine condition.
Hopefully, this short description of the auxiliary battery's purpose will save other owners the trouble and expense of trying to figure it out on their own.

In the middle of a stop-start cycle, the auxiliary battery won't kick in to get the car going again. You can't even start the engine once with a 12-volt 12-13Ah battery, much less do it multiple times.
Cranking the engine is the exclusive responsibility of the main 12-volt 70-80Ah battery.

The car's auxiliary features, like the entertainment system, climate control, lights, etc., get their power from the auxiliary battery.
This keeps the main battery from getting completely dead during lengthy traffic hold-ups or other situations where the battery life would be severely compromised, making it impossible to start the engine again.

Because of this, the ECO function is disabled in order to protect the main battery's ability to crank the engine in the event that either the battery's power is insufficient or none of the other conditions are satisfied.
After I changed the broken vacuum switch on the brake servo, my ECO system started working again. It fits perfectly, and I used a pattern part that cost only £25 to make it.
The strangest part is that whenever I start the car, I disable the Stop-Start function because I hate it. I hated it anyway, but the fact that it wasn't effective was even worse.

My sincere wish is that that clarifies everything.

It's not true that a car couldn't be started with a 12–13 Ah battery. Electric vehicle power surge protectors accomplish this.

A starter would only require a couple of seconds of current greater than what a 12-amp-hour battery could supply. Actually, it wouldn't be able to do this over many cycles, but it can move packs.

I appreciate you getting back to me. My original intention in writing this post was to dispel any confusion regarding the auxiliary battery's role in fixing the ECO Stop/Start feature that wasn't functioning. Your criticism of my attempt at assistance is deeply disappointing.
On the other hand, a motorcycle or tiny engine might get going on a 12–13 Ah (amp-hour) battery, but a regular car probably won't.
You need a high 300-600 Cold Cranking Amps (CCA), not a lot of capacity, to start a car, even though a 12V battery has the correct voltage. Between 500 and 800 CCA is the usual range for a 50-70Ah automotive battery.
A 12–13 Ah battery will have a CCA of about 200 and is intended for low-current, long-duration use rather than high-current, short-duration starting.
When subjected to a significant amount of current, a small battery's voltage rapidly drops below 8-9 volts, rendering it incapable of powering the vehicle's computer systems or fuel pump.
However, portable Lithium jump starters are designed with high discharge CCA rates to deliver high amperage for a few seconds, even though they typically have small Ah ratings (around 10-20 Ah).
The performance of the Lithium battery in a jump-start pack is superior to and requires less power than the auxiliary AGM battery that comes standard with our vehicles.

It was never meant to start anything, and the tiny wires on mine (and I'm sure they're on most of them) simply couldn't handle the current needed to crank my V6 derv. Mine has one, but it doesn't have stop-start. You wouldn't be able to shift into neutral and go if the main battery died, but it does have an electric column change gearbox. Among other things, the auxiliary battery ensures that the HOLD function continues to function, which is what makes this possible......By the way, unlike many others, mine is 1.2 Ah, not 12 Ah.
I could get by with a 12ah battery because my starter motor only needs about 100 amps, but a real battery would be able to provide 800 to 1000 cca. It won't take long to drain a 12ah battery to 100a, and repeated draining will cause irreparable damage.

It seems like you're not getting my post.

You can generate a substantially higher current for a short period of time using 12 Amps for an hour.

For example, 12Ah is equivalent to 24 amps for 30 minutes, 48 amps for 15 minutes, 96 amps for 7.5 minutes, and so on.

How much juice can a little jump pack hold? Such battery packs are able to start engines due to their extremely short but high transient current capacity. Small batteries with a high amperage rating can start engines, though the power delivery isn't strictly linear.

I have started 3.2-liter diesel engines and 3-liter petrol engines without completely draining my jump pack.
The deciding factor here is energy density.

However, it is unable to accomplish this because it is not linked to the appropriate circuits. Take the main battery out and try to crank it; it won't.

My post was about how a 12Ah battery, when connected to the main battery like a jump pack, has enough power to start a car—not that the MB auxiliary battery can do it.

This isn't entirely accurate. To begin, how do you suppose the backup battery would fare if the primary battery were to die from a series of stops and starts? When the engine is turned on, both batteries are charged, but the main battery has a far higher energy-to-time ratio. Although the remaining total energy for starting would be the same, it would be more cost-effective and efficient to install a main battery with a capacity of 70 + 12 Ah rather than one with a capacity of 70 plus a small 12 Ah battery.

Your list of consumers is correct, but the point is that the stop/start battery is only supplying those consumers while the starter motor is operated (plus margin). Customers won't notice the start-motor-induced voltage spike or dip, which won't disrupt music or other electronics.

It would be helpful if someone could list the various stop/start methods used by Mercedes, comparing them to the additional 12 Ah battery, and explain the differences between them.

You can get an alarm battery, which is the same size and has the same connectors, from any decent electrical wholesaler for around £12.