> So it's clear how many times more capacity a 20100mAh power bank battery has over 5000mAh phone battery

We don't, because we don't know the pack's battery voltage nor do we know the phone's battery voltage. You're assuming they're both 3.7V, but its not listed so we don't actually know unless we tear it open. In fact, my phone is rated for 3.85V, if the pack was rated at like 3.6V or less that's a decent bit of difference. Over 20,000mAh with that voltage difference we're talking about a discrepancy of 5Wh, which for a device using about a watt means five hours of difference in run time from that tiny discrepancy. Which, as mentioned, would be a good bit closer to a realistic answer if it just listed it in Wh, because personally I don't care what its internal pack voltage is I just care about how long it'll keep my 1W device alive. And even then, theoretically this battery pack can charge my laptop as well, that laptop's battery is definitely not even close to the same voltage as that battery pack. So comparing Ah to Ah there is pretty meaningless.

I've got a battery pack here rated for 5,000mAh. Roughly the same size as my phone, huh. Oh wait, its 56V. So roughly how many times can I charge my phone's battery off this pack? Going by Ah, once! But if I just knew my phone battery was ~17Wh, and we knew the battery was 280Wh, then we can guess we'll get about 16ish charges off of it.

https://www.jackery.com/products/explorer-1000-portable-powe...

Would you honestly state for understanding the capacity of the above device it would be more useful if it gave some rating in purely Ah? We've got no info on its internal pack voltage, that's practically useless to us. Its input is 24V, but is that what its internal pack actually is? Who knows. Clearly 3.7, because that's what we'll assume every pack is! Then its running things at all kinds of voltages: 12V, probably half a dozen profiles of USB-PD, and 110V.

I want to charge my car and I'd like to know how long it'll take. I'll do the back conversions to go away from the useful units here so we're talking just purely amps here, and its a 170Ah battery pack. I'll be pulling 40A from the wall. So roughly four and a quarter hours to charge, right? Nope, we didn't normalize for voltage. Well, we can normalize for voltage if we're just looking at watts. So its 240V 40A = 9,600W. Its a 68,000Wh battery pack. So the answer is a bit over 7 hours.

And sure, you're absolutely right about c-rates. The total useful capacity of the battery is going to be different depending on the current you draw on it, fully agreed. But that same logic still applies if you're talking about runtime and you're doing your math purely in Ah. Running at half or double the amps doesn't necessarily mean you'll get twice or half the life, your unit of measure doesn't change that, so I don't get why you're bringing up concepts like wire thickness when discussing which is the better unit to measure battery capacity in a general sense.

Sure, if we're talking purely about you building a circuit and wanting a battery matched to the circuit exactly, matching the voltage is useful. And when you actually do have voltage fixed, Ah becomes relevant. But in the real world, voltages are all over the place. You don't know what my battery pack here is rated for, you only have a guess its 3.7V. That assumed 3.7V is different from the 3.85 my phone's battery is rated for and is way off what my laptop is, so its not a straight 1 to 1 comparison.