Yet it does. They must be bound to serve as Congresspersons, it is clear. According to the rules of grammar, one cannot be bound without first taking an oath or affirmation. It doesn't say "will be" it uses the past tense for a reason.The document says nothing about the point in time at which they shall be bound. As such, it doesn't matter when it happens. By contrast, the President shall be bound prior to entering office.
I'm good with your attempt to tweak your way out using actual language in the constitution, it is where we find that there is no requirement as to where they take the oath, or which ceremony. The problem is you have to actually pretend you don't understand English to make it plausible.
And again, it doesn't say where they have to take the oath, they were constitutionally sound (took the oath), just not congressionally sound (didn't take it where they should according to congressional rules).