I like your reason for not getting 2 at the same time.
Even though I may be wrong about this, since an immediate immune response happens when vaccinated, I've wondered if getting 2 vaccines at the same time might either make the immune system work too hard, or perhaps just not as robust since it's spread too thin. I'm definitely not an expert on the matter of vaccines and immune responses, but it just seems like that could be a possibility. Hence me not wanting to double up on vaccines.
And I especially wouldn't want to for the high-dose version on the flu shot as that seems like a double dose in itself. And with a Covid shot added on top of it, it just seems like my immune system could be overworked, or spread too thin, which wouldn't help it build antibodies.
All MOO. (yes, I know, I think too much. lol)