Gmail web says “not sent to spam”, downloaded version doesn’t

I would let one account handle the spam filtering for all of your accounts. I don't believe there are many benefits to having each account handle its own spam. I think Gmail is aware of the destination address of each incoming mail and having identical spam emails (with only the destination address changing) will strengthen the spam algorithm just as much as if it were single emails sent to multiple accounts.

Regardless of how much (more) it might improve the spam algorithm, you should choose the solution which best improves your workflow. I would just go with the first option (spam filter on the main account).

FYI: I use Gmail to process email from 12 accounts across 7 domains each with spam being primarily handled by my main Gmail account. :)

If a message matches your filters, those will always take precedence, I believe.

That's why you can also opt to "mark it as Non-spam".