In my algebra book my author does the same thing. Half of the discussion is on showing the set can be contained in some other set. But back then I did not see the purpose in doing that and just ignored that (in fact I once posed a question on this forum of my version of the proof of algebraic closure asking whether it is correct. Now I realize that it was not because I did not prove my set was countained).
In my book, Hungerford's "Algebra", his argument for algebraic closures is indeed very reminiscent of your argument. However, as he states before he goes into the proof,