I don't know much Japanese but I don't see it as that weird. I see it as something like "murder of crows" or "pile of sand". Something cultural that was there for a reason (or a monk somewhere) and now we have to memorize it.
This is what they are comparing to the difference between "first" and "one" in English, which are obviously two different origin words for the number 1 (unlike sixth and six, where sixth is clearly just derived from six).
We also have a LOT of these for plurals and take great pride in knowing all the stupid ones - "a gaggle of geese", "a murder of crows", "a herd of cows", "a fleet of ships". These are all really just the same as the measure words in Chinese with many specialisations for large groups of specific things.
We also have plenty of measure words in English when we take something from the realm of uncountable to countable, as I showed in the examples above, and in many ways Chinese tends to have less weird single-use ones like "gaggle of geese", just a larger variety of the common ones. So there's a special word for "long thin things", "vertical long thing things", "flat things on a (mathematical) plane", "a lump", "a slice", "a vehicle", "a machine" as well as the fallback 个/個 when none of the more specific ones are appropriate. I think it's only because we generally don't use any measure word for small groups of countable nouns in English that we think it's somehow strange that other languages do.
There's also not so many counter words to be a problem. Sure, the absolute number is high (some say very precise numbers like 214, 232, etc, some people low ball it at 100, some say over 900 words if you count all the old obscure words), but in practice, about a dozen of them will cover almost all the cases you think are hard for English speakers. So, 个，条，只，支，本，件，张, 家，座，部，辆，头 are probably the most common ones that don't usually have an English translation, almost every other measure word would also require a measure word in English, e.g. 条=strip, 张=sheet, 些=few, 双=pair, 样=kind, 种=type, 头=head, 片=slice, 分/份=piece, 点=bit, 块=lump, etc.
As with most things about learning a new language, it's also important to embrace the differences - they give you a new insight into how other people think about things, and seeing when someone else's internal model for something differs to yours can help you better understand your own model as well as the new one you're learning.
[1] You'll understand this if you ever read The Famous Five!
