Not necessarily, as the trained model can just be the matrix. It's more like data, as presumably the same algorithm with different weights, trained by different data would be permissible.
ML training is an algorithm that produces an algorithm as output
It doesn't really though. The prediction algorithm already existed before training, it just is a particular collection of parameters that the training produces. You wouldn't say changing the interest rate changes the algorithm for calculating interest payments. The weights can be viewed another input to the algorithm. You aren't going to outlaw linear regression, but a particular set of coefficients.