The presence of accidentals does not make a key. There is no “Mixolydian” key.
Key signatures and accidentals are tools to make sight reading easier. If you’re in the key of Bb, and you come across a note on the third space up from the bottom, you know it’s an Eb, unless there’s a natural accidental making it an E for the rest of that bar.
These aren’t laws, they’re tools, but the tools are only useful if everybody using them understands how they work.
Arguing that a key signature isn’t a key seems outside the useful range of these tools.
Sure, you *can* hang drywall with framing nails, but if your buddy comes over with a screw gun and a box of drywall screws to help, and you insist he hammer it up with framing nails, he’s gonna leave. Same with me and anyone who insists a key is in “F Mixolydian”. No, that right there is the key of Bb, and that right there is an Eb with no accidental, and that right there is a Dominant pedal, and this right here is me packing up my stuff and leaving.