I think there is a strong belief in the people who comment here that proof skills are more important than calculational skills. But I am not so clear that this is really true, is not completely how the human mind works.
I mean does understanding Dedekind cuts (or whatever they are called) really help you do arithmetic? If you learned that and never learned long division, versus someone who did the converse, who would be more powerful if you had to calculate something?
Some manipulative skill can be very helpful when doing trials or developing new mathematics. And the algorithms and methods developed over the years for arithmetic, algebra, and calculus have a lot of power in them. Represent a lot of distilled experience and effort to make things clear and simple and useful.
Ideally you should have both. Understanding of concepts and manipulative skill. And move back and forth between them. And have them inform each other.
Remember there a lot more people who need to understand math like Feynman did so they can follow physics derivations or do engineering problems than there are who need to play Bourbaki. And even for the Bourbaki soldiers, some combat with manipulation will make them better mathematicians.
This doesn't mean there is no value to concepts either. They can help with remembering methods, can help with understanding them, etc. But having a toolbag of tricks is valuable too. Don't look down on it.