The way I understood it was that it is supposed to be that people on the old chain can still use Bitcoin normally.
No, a hardfork requires
everyone to upgrade to a new client. After a hardfork old clients will outright reject the new chain.
If that does not happen, and if a significant portion of the hashpower refuses to upgrade, then Bitcoin will split into two completing cryptocurrenies that can co-exist forever. This is the worst-case scenario.