"the grid absolutely, positively, needs to supply current for any need that gets switched on"
Why?
A small addition to that could probably take us long way: Any need worth the cost.
If the price matches the supply I'm sure people will get quite creative in reducing the demand enough to make the problem go away.
It needs to because otherwise supply voltage drops and you get brownouts.
There is no provision in light switches, outlets, thermostats, industrial machinery, or anything else to first check "is there enough current in the grid right now to run this?". They just draw current when connected, and changing that policy would be a huge endeavor.
This is called "demand response" in utility circles, and to varying degrees, has been operational for decades. Lately it is becoming much more reliable and accessible thanks to smart grid networks.
If the price matches the supply I'm sure people will get quite creative in reducing the demand enough to make the problem go away.