I'm not entirely sure where you get the number 370 from. This website claims to have a list of 620 available to its subscribers, and I imagine that even that it not exhaustive.
It's not about sheer numbers though. Most verbs are regular, obviously. However many of the irregular verbs also happen to be very commonly used. Take a look at this list. It's the 10 most common verbs in the English language:
How many are irregular? How about all of them! Only 7 of the top 25 English verbs are regular.
Compare that to, say, German, which happens to be one of English's closest neighbours linguistically. In German there's only one truly irregular verb, sein (to be). Other than that, all verbs fall into two categories, strong and weak, which each have their own (different, but predictable) conjugation rules. Of the top ten German verbs, one is irregular, eight are strong, and one is weak.
Three forms to a verb? Regular ones have four (look, looks, looking, looked). Irregular ones have X forms, where X is a number perhaps less, perhaps more, or perhaps equal to four.
There are plenty of weird things in English other than spelling. How about...
The verb "to dust" can mean to add dust to a surface, but also to remove dust from a surface.
Irregular plurals: mouse/mice, goose/geese, man/men, etc. The plural form of "sheep" is "sheep".
For that matter, why on earth is there no singular for the word "cattle". They're a common enough animal, and it clearly makes sense to be able to talk about them in the singular. So why do we just have this plural word "cattle"?
There is of course "cow", but that refers exclusively to females. "Bull" refers to males, and "calf" (another irregular plural by the way, calves) to the young. There is no English word to refer to a single individual Bos primigenius without committing to its sex or age.
Given the two phrases "a cute little puppy" versus "a little cute puppy"; almost all native English speakers would agree that the second one sounds wrong. Virtually none of them could tell you why. :-)
Similarly, "a nice warm bath"/"a warm nice bath"; "a red Japanese car"/"a Japanese red car"; "an old woollen scarf"/"a woollen old scarf". In each case the first is right, the second is wrong.
Bags of synonyms. If you know what the word "moon" means, that doesn't help you if you hear the phrase "lunar cycles".
The main reason we have so many synonyms in English is that the language is a cut and shut combination of Low German and Norse French. So we ended up with one word inherited from German ("need") and one from French ("require") meaning exactly the same thing. We also ended up with a lot of (mostly technical) words derived directly from Latin without the French intermediary, and then to make matters worse the bloody Vikings gave us a whole bunch of other words we neither needed nor required.
(This also explains why nobody wins the human race. English has two words spelt "race" which are pronounced exactly the same. "Race" meaning ethnicity or population comes from French; "race" meaning a speed contest comes from Norse.)
And I don't want to bang on about them but... the idioms. Perhaps you and I don't see eye to eye on this, but as far as deciphering the meaning of a sentence goes, even if you know the definitions of the individual words, it's not a done deal.
On the other hand, perhaps I can see your point? After all, people learning English as a second language do seem to pick up the meaning of idioms as soon as they run into them. Maybe it's a piece of cake?!