Why do the reals need to be constructed? Do they somehow "span" the rationals, the roots, and the transcendentals like e and pi?

If you like, and some people do, you can forget about any construction of the reals from the rationals (or anything else) and instead define them axiomatically. One such axiomatization is Tarski's.

This approach will avoid any weird feeling you might have about a real number being an equivalence class of whatnot.

Usually, the reason to provide an explicit construction of something from a simpler things is that it proves that that something exists (mathematically). Moreover, it allows you to study properties of that something in terms of the simpler things that you presumably know better.

Nobody thinks of real numbers as equivalence classes of anything. Once the construction is done you can just forget about it if you like. Having a construction just means that the model of the real numbers that you fantasize about is at least as consistent as a model you might have of the simpler things. To some people it gives reassurance, to others a headache.

As for your attempt to define the read as something spanned by those things we have names for, together with some operations on there. The problem is that there are only countably many such things while there are uncountably many real numbers (at least if you believe that every real numbers admits at most two decimal representations). So this can't work. It might be strange to think about there being more reals then potential names or ways to approximate reals but it's a real fact (pardon the pun).

Constructing the reals is important if you want to do analysis. If you want to talk meaningfully about sequences or continuity, you need to fill in the "holes" in your space. You're coming from the perspective that we built the reals because we need "more stuff", but that's not the case. The reals are designed to fit together a certain way, and it just so happens that you need a lot of stuff to do that. If all the interesting analysis we wanted to do could be done with a smaller, countably infinite structure, it's possible that's what we'd call "the real line". In fact, I think some people do try and do analysis with the computable numbers.

You can get away as follows: You demand axiomatically that there exists a complete ordered field. It can be shown that any two such fields are canonically isomorphic and thus whatever someone assumes to be his personal idea or mental representation of $\mathbb R$, it is essentially the same as other people's idea as long as they agree to talk about a complete ordered field. (You than rather obtain $\mathbb Q$, $\mathbb Z$, $\mathbb N$ as subsets instead of constructing the other way round)