Here is a nice discussion of what happens when you don't validate your elliptic curve parameters properly:

* https://research.nccgroup.com/2021/11/18/an-illustrated-guid...

The highlight here is that in some cases, failure to properly validate gets an attacker the secret key material.

Note all the conditional bits. Different curves have different properties and different issues. There are a bunch of different curves in common use while RSA pretty much always uses the same value for the parameter these days (RSA literally has just one parameter. The exponent.).