The voltage range of aged batteries is the same as the day they leave the factory. Or it could be even larger if the BMS is set up to sacrifice a little nominal capacity für lower wear when new, slowly widening the range as capacity decays. The temporary voltage drop while drawing a large current will be bigger on an older battery, that much is true. But that would still be a smaller delta than the one between fully charged and not at any age. If there's performance difference, I'd expect it to be from deliberately yearning performance/runtime trade-offs (which arguably are better to have than not have)
If your Lithium Ion batteries are at 1.3V they are way beyond their service life.
Normal voltage range for Li Ion is 2.5 to 4.2V, nominal voltage is 3.7V, some manufacturers recommend you never discharge them lower than 3V. So something is off or you have your battery chemistries mixed up.
This is typically specified in open circuit voltage. Under load, lithium ion batteries can get to VERY low voltages due to internal resistance. 2.0V is not uncommon.
I've built a bunch of Lithium Ion packs (nothing huge, the largest was 10 KWh, but still you don't want to mess that up, that's 1000 cells in a 40S25P arrangement), none of them go that low under load. If your cells drop down to 2.0V under load that's not a normal condition, either you are discharging your cell at more than the 2C or so that is normally specc'd (which you are welcome to do but it will cost you in lifespan) or the cell is on the way out. The normal arrangement for high current discharge is to simply set up more cells in parallel so they all carry only a fraction of the current and stay well below the maximum permissible current.
One application where cells are loaded up really heavily is in RC toys and drones, there lifespan is secondary to performance. But a normal, long-life application for a Lithium Ion battery pack will ensure that batteries are not overloaded (either during charge or discharge).
There are also special cells that can gracefully handle high discharge current (and usually correspondingly high charge currents) typically at the price of some capacity for a given volume.
You'd be surprised the peak current demand of modern smartphones. AVERAGE current demand (not peak) of a Snapdragon 8 Gen 2 can hit 16W for short periods of time. At 3V, that's 5+ amps out of the battery. Worse if you account for efficiency, and even worse if you account for other power draws such as camera, display, etc etc. Running at 3-4C for short bursts is not out of the question.
That I readily believe. But in a well designed system (designed for longevity) you'd never see such discharge currents in a sustained manner and even short bursts are better served by capacitors (charged at a lower constant current) close to the consumer than by batteries. And that's why you'll see voltage regulators and large banks of capacitors right next to the CPU (besides compensation for line losses and stability of the supply, which with rapid wide load variations would cause all kinds of problems for the logic).
But in for instance a vehicle or an e-bike you'd rarely see batteries drained faster than the spec, not if you want to use your expensive device for a while.
Keep in mind, when I say 'short bursts', I mean on the time scale of seconds. No capacitor bank is going to be capable of supplying current for that long; especially not one put in a phone. In most cases, the restriction is inevitably heat based (how long can I run this before thermals cause me to need to throttle) and the power supply is pushed to be able to handle it.
EV batteries and e-bikes are a whole different animal than what is used in cell phone battery tech.
Also, is 2.0V bad for the battery? It's not great, but it's not Voc of 2.0V. It's Vsc, essentially. Which is a very different quantity.
