As a fellow college algebra (et. al.) instructor, I think that knowing specific methods is a necessary thing to confirm. That's what I do on my tests. If you want and have time to add yet another general question (so as to assess both), then that's commendable.
The point is that the specific methods will pop up in future courses in different contexts, and your students need to be proficient enough with them to follow the development. This could be in the context of more advanced definitions, proofs, algorithms, etc. Here are some use-cases off the top of my head (maybe others can come up with better ones):
Method of factoring: You don't mention that the Fundamental Theorem of Algebra is part of your course. At some point math students should digest this, and factoring is critical to their awareness. Ultimately it leads to the fact that any polynomial of degree $N$ can be factored into exactly $N$ factors, a wonderfully clear observation. Factoring will be used to reduce lots of stuff in calculus, so getting more practice there is good.
Solve by square roots: Students should be aware that solving an equation by square roots gives two solutions. This prepares them for higher radicals in the complex plane; cube-roots of unity have three values, etc. And it's needed as part of completing the square. (Also personally I feel it's by far the simplest method, and if a technical person can't do this mentally they'll seem slow.)
Completing the square: The essential thing is that this is the method by which you'll prove the validity of the quadratic formula. (You do that in class, right?) With this in hand, I find it's actually a fun class moment to see how much the students can help drive that symbolic proof. This activity paves the way for more proof-based courses in the future; you should verify that students have the tools to follow that proof, say. Another application: manipulating general circle equations so as to easily graph them.
In some sense, the quadratic formula is actually the most limited of all the techniques you've listed, because -- while formulas exist for 3rd and 4th degree equations -- a major achievement in algebra was proving that no such formula can exist for equations of the 5th degree in higher. So solving by a formula is ultimately a dead-end. In contrast, at least hypothetically, any such equation still has a predictable number of roots, and can always be factored.
IMO: It's easy to lose sight of the fact that almost everything we do in a math course like this is an exercise, not really a problem. (And I'm careful to use that former terminology.) Does anyone in the real world need to literally solve a quadratic equation by hand? Well, no: you can just use some technology (Wolfram Alpha comes to mind.)
The real point of our math classes is to understand concepts; like, any nonlinear polynomial is factorable (in complex numbers), that $n$th roots have (you know) $n$ roots, and that we need to verify our truths through proof (such as the techniques of completing the square vis-a-vis the quadratic formula). The test questions we present are merely somewhat-regrettable verifications that the students have indeed absorbed the mathematical facts.
So again on my test at this point I ask specific questions directing that students show me they know how factoring, completing the square, and the quadratic formula work. Separately, a homework quiz has questions on solving by roots, the key step in completing the square, and recreating one step in the proof of the quadratic formula.
If you add another general question so that you've also assessed their general ability, great -- but you will probably find they just use the quadratic formula, so you're not getting much extra information from that. Personally I might lean more towards a written question a la, "Given the following equation, which method would be simplest and why?", but you might get a bunch of pushback on that (and I don't have space on my own tests for it).
