Typically you fully charge them, light goes green, then start to use it. After about a minute or two the cleaner cuts out. But you can continue to keep switching it on for ages, 20 minutes or so, provided all your cleaning can be done in 10 second chunks. Why does it do this?
Broken-by-design, by one way or another. Likely some kind of protection feature, which has been tuned wrong. I wouldn't even rule out planned obsolescense. Older battery technologies were even more crappy, so people are mentally still accepting that "batteries just die".
If the battery is old and reaching its end-of-life (years of use), its DC resistance has increased, which the thing would detect as an excessive voltage drop. But if you really can go on with this for 20 minutes, it seems it's a badly mistuned protection. If the battery actually was that bad, you wouldn't be able to go on for that long.
The question is, how do you get equivalent performance to a corded cleaner, taking 600-800W, from a battery which is from 22V to 40V?
First, the voltage doesn't matter, naturally; it's the power. In principle, the same power can be delivered at 20V, 200V, or 2000V. (Of course, extreme ends are impractical.)
But, they also do have much lower power consumption (< 300W) than the corded cleaners (ranging typically from 800 to 1600 W).
Their absolute trick #1 are the rotating thingies at the nozzle! Using little motor power, these physically remove the dirt (embedded into textiles, for example), and once it is up in the air flying, the vacuum can pick it up way more easily.
All the rest, such as using BLDC motors and making a big deal about it, is small optimization.
The weight is around 2.2kg so battery weight can't be more than, what, 30% of that?
Assuming the battery would be 1kg, and using modern power tool cells, this would equate to around 200Wh capacity, approx. $50 raw cell BOM cost. For the 40 minute runtime specified, the power would be around 300W.
The Lithium batteries used are, presumably, AA size since other battery powered things that have failed seem to use that size cell. Just how much current can you take from an AA cell? For how long? How many recharge cycles will it do?
They use 18650 cells (18mm diameter, 65mm length), which is much bigger than AA.
A typical "power tool" type of 18650 cell easily supplies (at acceptable efficiency) continuous current of 4-5 times it's capacity, meaning it can be fully discharged in about 15 minutes, if necessary. For a typical 2.5Ah power tool cell, this means over 10A. Cells can be paralleled to increase the max current, and put in series to increase the voltage. (The best ones are specified to 30A discharge, but this is only possible given brand new cells with no aging-induced increase in DCR, and still tends to mean case temperature of 80-90 degC at the end of discharge, definitely not recommended.)
Typical 18650 li-ion cells do about 500 to 2000 cycles depending on how hard they are pushed, how deep the cycles are, etc. 500 cycles is about right for fairly hard, yet non-abusive full cycles. End-of-life tends to be defined as a drop of capacity by 30%, or increase of the resistance by 100%, whichever comes first.
Note that the doubled resistance already makes most power tools quite crappy if there was no proper "extra power" available initially. This is actually what can make a huge deal in actual battery life. Say, your vacuum or drill
needs 200W to be usable. Now, design a battery pack to provide 300W max when new, and it's total crap after a year when the resistance has only risen by some 30-40%. But, design a slightly bigger pack which can supply 500W, and is charged at a slightly lower C rate (although the actual charge power is still the same), and it may be usable for over 5 years just fine!
Whilst the cleaners have guarantees, the batteries seem to be hedged by exclusions.
This sucks. Li-ion systems can be designed to last fairly well, but such exclusions do not motivate the designers to do good job.