A degraded battery can't deliver as much power as a new battery and for your phone to feel fast you have big cores to boost to over 3 GHz which requires a relatively high voltage. These voltages aren't possible with a degraded battery and that's why your phone feels slower.
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.
