Firstly, if I'm correct in understanding your question, you are asking why one needs to regulate field theories since we renormalize them anyway?
Assuming this, here's my understanding:
Quantum field theory in general has several divergences, which we must deal with. We get around such divergences by dealing with renormalized quantities. But how does one renormalize the theory? Essentially (without going into too much details), the statement is that quantities in QFT often have infinities. However, if we calculate the "right" quantity (one has to be more specific about what is right and what is not, that is a slightly harder question to answer), the infinities cancel. But one cannot simply say $\infty - \infty = \text{finite}$. That makes no sense. One has to describe the precise form of the divergence. This implies making the $\infty$ more precise, by writing it as $\infty = \lim\limits_{\Lambda \to \infty} \log \Lambda$ or $\infty = \lim\limits_{\epsilon \to 0} \frac{2}{\epsilon}$, etc. The process of characterizing the infinities in this way is regularization. Once you have done this, the expression $\infty - \infty$ now makes complete sense.
We see that regularization is an essential process in renormalizing a theory. Good!
(Let make another comment about the symmetry statement you made in your question) The problem with introducing regulators is that often the regulator used may not satisfy all the symmetries of your theory, in which case we have quantum anomalies. One has to deal those after regulating the theory.
With regards to your comment on counterterms surprisingly cancelling divergences, I will say this: It is not all that surprising. In fact, counterterms are introduced by hand in the action specifically to cancel all those divergences. However, what may be surprising is that in some theories (called renormalizable theories) a finite number of counterterms are all that you need to cancel out all the divergences and that all the divergences can be swept under the rug by redefining your bare fields and couplings. While this is a little bit non-trivial to see, a simply dimension counting of divergences makes it easy to see (atleast at 1-loop level). For most other theories (non-renormalizable) there aren't a finite number of counterterms and one has to modify the action by introducing an infinite number of counterterms and operators.
However, technical discussion aside, if QFT is to be a good theory of nature, one MUST get finite answers out of it and there MUST be enough counterterms to cancel all the divergences. Either this happens on its own (renormalizable theories) or we have to impose it as a condition and modify our theory to accomodate all the divergences (non-renormalizable theories)
