Suppose I have some function $F$ that can be written as an expression $A$, and that $A$ can be approximated as $B$, and that $B$ can be rearranged to give $C$. While demonstrating the derivation for $F\approx C$, one might write this out as: \begin{align} F &= A \\ &\approx B \\ &= C \end{align}
However, someone reading this without much attention might incorrectly interpret it to mean $F=C$, especially if there are many more than just three lines - they might miss the approximation.
An alternative approach would be for the approximation to persist going forward: \begin{align} F &= A \\ &\approx B \\ &\approx C \end{align} But this has the disadvantage of giving the impression that $C$ is an approximation of $B$ rather than a direct rearrangement.
Of these two approaches, is one more common than the other? Can you make a general case for favouring one over the other?
The first form is more common; the second seems confusing. Someone reading without much attention might draw all sorts of incorrect conclusions; I wouldn’t see it as a central task of notation to guard against this. As you rightly say, the inattentive reader might draw incorrect conclusions from the second form, too.
The decisive advantage of the first form is that the attentive reader has precise information which relations are equalities and which are approximations, which is not the case in the second form. There would have to be a very strong case for improving the understanding of the inattentive reader offset this, and you rightly don’t claim that there is.