How to deliver an encrypted message to multiple recipients without revealing keys
Also, if I give people the private key, and I keep the public key safe, am I right in believing that it's possible to generate a new public key just using the private key itself?


Correct. The private key contains the modulus, one or more of the primes, the private exponent and often some other relevant things. The public key is just the modulus and the public exponent, which is usually a small number, often 65537, and can be calculated with the private key even if it isn't.



What you actually need for your problem is authentication. You sign the data (e.g. with RSA-PSS), which you may have encrypted (whether with a symmetric key that you give the recipients or their public keys). The data can even be public if its secrecy does not matter.

The recipients can then use your public key (which they must somehow get first) to verify that the data was signed with your private key. Assuming you do not reveal your key, no one else can forge a valid signature.