It’s not so much the AMOUNT of rain as the TYPE of rain. April tends to have a a lot of convective showers, because there is often low pressure (due to the jet stream movement you mention) and the sun is strong enough to trigger shower cloud formation.
So, rather than the drizzle of winter, you tend to get sunny intervals interrupted by sudden downpours.
No, but there was a she who sold seashells by the seashore.
By March, our continuous heavy Winter rains have usually given over to our continuous heavy Spring rains …
If winter comes, can spring be far behind?
There are lots and lots of folklore sayings about weather, I would imagine in most cultures, and they must just have seemed generally true a lot of the time; so, in England, “February Fill-Dyke” and “March - in like a lion and out like a lamb”. But they must also be fairly dependent on place, and not necessarily translatable to the places where the people who grew up with such sayings emigrated to.