If you include applying axioms or transformations as part of derivations or proofs as calculations then, yes, you should get good at doing them. If one is poor at copying long expressions with the proper substitutions then there is a modern answer. That is to use an application such as Mathematica (the one I'm familiar with and reference, but there are others).
The trick is to replace one skill, error-free (and neat) copying of long expressions with substitution or transformation, with another skill, writing rules and routines that specify the axioms and transformations and applying them. I argue that the second skill may not be faster to use, but it is less error-prone and more concentrated on the actual mathematics.
A second trick, when using Mathematica, is not to regard it as a scratchpad, or a super graphical calculator, or a programming worksheet, or a mathematical word processor (although it is in part all of these things), but as a blank sheet of paper on which you are developing/learning and writing your mathematics. It could, should actually, contain sectional organization and textual description and discussion as well as calculations. It is a rather magical sheet of paper because it has memory, can do active calculation, can accumulate knowledge (in the definitions, rules and routines you specify), can contain beautiful graphics and dynamic displays of various types.
The next trick is to make a sincere attempt to calculate everything. Don't fill in with word processing. I don't claim, or am not certain, that this can always be done but a pretty wide and deep swath of mathematics can be done by computer calculation. This has several advantages: the entire calculation, derivation or proof is largely self-proofed in the sense that the calculations won't work with input errors; the starting point (axioms, theorems, and transformations used) must be present, sometimes a confusing issue with students; the gap between the starting point and the desired result is more clearly defined for a researcher or student.
Like any computer document one can revise and edit. You could have Try 1, Try 2, etc., in separate sections and then throw away the failed tries. You don't have to keep copying the starting expression by hand. The finished document is useful and can be referred to in the future or added to. Knowledge accumulated could be passed to other notebooks or upward to packages used by other notebook documents. There are tremendous advantages in this approach.
The disadvantages are that you do need to have the Mathematica application and there is an extended learning curve. It is unlikely that you could buy it and then start using it off the self for any significant mathematical problem.