I think there's no good reason to insist on $\iff$. First of all the obvious: $p\iff q$ always implies $p\implies q$, but not the other way around. Since beginning students may not have that totally nailed down yet it's better for them to err on the side of $\implies$ rather than mistakenly write $\iff$.
Now about extraneous solutions, I think that technique wouldn't help much anyway because the point is that if you square the equation and get two solutions $x_1$ and $x_2$ you still need to check which of them satisfies the original equation, whether you are aware of the "broken" $\iff$ relation or not. So actually you are more likely to check if you are aware that it is $\implies$ all the way through, since it is a reminder that the equation implies the solutions, but not the other way around.
In other words, I think what's more important is to understand that when we solve an equation, we are interested in the solution, and therefore the direction of $\implies$ is what interests us, that is, we are interested in making a series of logical steps that imply one or more possible solutions. For that exact same reason, we need to check which of our solutions satisfy the equation if we get more than a single one.
Of course, at the more advanced stage the student would know ahead of time how many solutions to expect, but I'm assuming the question is not concerned with that level of math yet. Although even at an early stage, it is quite easy to explain to the student that an equation of integer degree $k$, that is containing $x^k$ can have at most $k$ solutions - this can just be given as a simple "axiom" at first, with a promise to be explained more deeply some time later...
Edit: After thinking more about this, I find it important to add that as we know, it's taken for granted that we don't teach propositional logic at an early age. I'm not sure if we should teach this. (At that age, it is certainly in the area of "experimental education" as far as I know). However, if it happens that the young students are taught propositional logic, then it recasts the question in a totally different light. In that case, I would say it would make much more sense to expect the student to be able to distinguish $\iff$ steps from $\implies$ steps, and that it would in general aid them, not only in algebra equations but also in all areas of math. So perhaps this question of whether and how to introduce propositional logic early on, is one that should precede questions about the usage of propositional logic.