I think this is a case where there used to be a distinction between the two phrasings in English... but the distinction is archaic, and the use of As in a So situation would be contested largely by older people that actively study prescriptive grammar. Language is always changing.
I didn't know the answer so I checked my dictionary and examples it has. I did find an example when tanto would be 'as much' and that was when in the context of amount or quantity. So you aren't wrong. As for 'no tan' I can't find any reference to support using it in this context. Hope it helps you, it did for me.
wait... ok, so for tanto to mean "both" it has to be used in tandem with "como". And when used together, "como" replaces what would normally be an "y"
i.e. hombres y mujeres becomes hombres como mujeres if tanto is used in the beginning.... which creates a "both" statement.
Sino is an elegant way to provide a non-obvious alternative to a negative statement, without having to start a new sentence.
To avoid confusing it with pero, try replacing it with although–if it works, it's pero; otherwise, it's sino.
> — Me apetecía verte hoy, pero hasta mañana no puedo.
> "I felt like seeing you today, (but/although) until tomorrow, I can't."
> — Pues yo no contaba con verte hoy, sino mañana, así que perfecto.
> "(Well) I wasn't counting on seeing you today, (but/I was counting on seeing you) tomorrow, so that's perfect."
I go deep into the usage context of sino in this NachoTime post.