Why the "find things that add and multiply" is taught before the quadratic formula
This method of "think of $p$ and $q$ so that $b=p+q$ and $c = pq$" is used in the service of factorising the polynomial equation into $(x+p)(x+q) = 0$. The concept that you can factorise and then set each part equal to zero is the fundamental idea of this method, and is in fact a fundamental idea to solving many equations, not just polynomial equations. As such, it's a part of a larger more general method that the quadratic formula is not part of. Often this section of the maths curriculum follows a section where you tell them about noticing a common factor (eg in $2x^2 + 4x = 0$ becomes $2x(x+2) = 0$), and is trying to match with this earlier idea.
The quadratic formula is usually introduced as a "last resort" when all other methods have failed. Personally, I have a sense of power when I think of a way to factorise it, rather than "resorting" to the quadratic formula. I conjecture that many people feel that using factorisation is more "righteous" than using the formula, in the same way that they feel guilty asking a computer to solve an equation for them. While you can argue on the correctness of this feeling of guilt, you can't deny it's there for a lot of people, and many of these people are teaching the students. It is probably worth noting at this point that we don't teach a "linear formula", that solves $ax + b =0$ by doing $x = -b/a$, because there are other more general methods that work just fine.
Another reason is that the concept of writing a polynomial as a product of factors is an important idea even if you are not trying to solve equations. As algebraic objects, polynomials have products and factors the way that integers do, and factoring them is a way of reducing them to their "lowest form". The quadratic formula is a way of solving an equation and is rarely used as a way to find a factorised form. Of course, there is nothing stopping you doing so -- I do find it odd that it is rare to see someone saying that they found $x = p, q$ by the quadratic formula and therefore the factorised form is $(x-p)(x-q)$!
Factors are also a way to strengthen the connection between a polynomial and its graph. Noting that you can tell the x-intercepts of a function from its formula when it is in factored form is a powerful thing. It's cool that numbers pulled out of the structure of an equation written on the page tell you about what the graph is doing. Interestingly, completing the square also strengthens this connection, because when you rewrite $y = ax^2 + bx + c$ as $y = a(x-d)^2 + e$, then you can immediately read off the location of the vertex $(d,e)$. Unfortunatley, I find it rare that students know that completing the square allows you to do this trick. (Note that we do often tell them a bit about this with linear graphs, citing the $y=mx+c$ and the $ax + by = k$ forms of a line that tell you different information.)
Why not teach (synthetic) polynomial division
I never learned synthetic division, and when I saw my South Australian friends do it when I got to Uni I found the whole thing a bit mystifying. (I may have been biased, since I already knew how to do ordinary polynomial division it seemed like a waste of time to learn another method when I had one that worked well for me, especially considering how few polynomials I really needed to divide even in my first year of a maths degree.) When I look at the first few hits of a Google search for "synthetic polynomial division", the explanations there are not particularly illuminating. It is not at all obvious how synthetic division might be used to extend understanding from quadratics to higher-degree polynomials, except for dividing by linear factors until you get a quadratic, which ordinary polynomial division does too.
So perhaps the question is really why is polynomial division (synthetic or otherwise) not taught earlier? I wish I knew. I think it would be great if the process of dividing polynomials was taught earlier. As algebraic objects it makes sense that if you can multiply polynomials then you would want to divide them too, and also it has a nice lead-in to rational functions. As you say, it would allow extension of quadratics to other functions.