Just buy the xbox controller then.
And you expect to pair that controller with a PS4/PS5 then? How about a Nintendo Switch? Right. Let that sink in a moment.
I genuinely don’t understand
That statement really says it all then doesn't it, very clearly. Allow me to sum it up for you. The batteries in most controllers are not easily replaced(Xbox controllers being the exception). They only have a limited number of charge/discharge cycles in them so every time they deplete and recharge, that is one less life cycle that battery has in it. And even the most high quality batteries lose capacity as they wear out. So given that simple understanding, I(and every other gamer with more than two brain cells to rub together) would like to have the longest life possible out of the batteries supplied with our controllers. This means longer lasting individual charge cycles, which means more gaming inbetween charges and thus not needing to charge said battery as often, therefore making the bloody thing last as long as possible for maximum controller longevity.
In order to determine the size, cost and weight effects of a bigger battery you first need to state the desired capacity.
The standard DS4 controller battery is a 1000mah unit. There are aftermarket replacements that have upto 2500mah, but using them voids the warranty(not really a problem after 90days). I have done this to all four of my DS4 controllers and the replacements fit right in. The original batteries started having problems just after the first year of use. The replacements are still going strong after 3 years of use. Imagine that. The replacement batteries also provide around 20hrs(still) of playtime per charge. To sum up, a little added weight that isn't noticed, more than double playtime between charges which results in longer over-all battery lifetime. The bigger the battery the better. Lithium Ion batteries don't weigh very much anyway.