I had a thought about this today. Completing a space is a bit like getting a bank loan to buy something really nice (bear with me on this). Because:
If you have enough money already, getting a bank loan isn't worth the hassle.
If you can do your analysis without needing completion, then it's simpler to just do it.
The loan still has to be paid back, but having a loan means that you put off paying until later.
As others have said, a common use of completion is to use Hilbert space techniques to study non-Hilbert spaces. But often one wants to know that the final result is in the original space. So using Hilbert space techniques is a way of putting off questions of existence until later.
If you're a financial wizard, you can take out the load, use the money to make more money, repay the original loan and end up ahead of the game.
Sometimes, just sometimes, once you've done the Hilbert space stuff then all the rest just falls in to place.
The point, such as it is, is that when you have an incomplete Hilbert space then completing it adds in stuff that you didn't want - if you did want it then you would have put it there to begin with. Thus when we complete continuous functions to square-integrable ones, we do so in the knowledge that we'd really rather be using continuous functions as they're much better behaved than these nasty not-quite-functions.
John Baez is fond of a quotation attributed to Grothendieck: "It's better to work in a nice category with nasty objects than in a nasty category with nice objects.". One could adapt that to Hilbert spaces: "It's better to work in a nice vector space with nasty elements than in a nasty vector space with nice elements.". In this respect, Schwarz functions are some of the nicest functions you could met, but they live in student accommodation. On the other hand, square-integrable functions have some undesirable personal habits but live in a fantastic mansion.
And to underline my last point, sometimes it's possible to go to a party hosted by the Square Integrables in their posh mansion, but spend the whole time hanging out with the Schwarz family.