The "Pickup Group(s)" field in the extension's configuration identifies one or more call groups from which the extension is allowed to pick up. The call group to which the extension belongs is specified in the "Call Group" field.
It sounds like you configured all extensions to have "1" in the "Pickup Group(s)" field (that means all extensions will be able to invoke the Group Pickup feature - or F75 - for extensions that belong to call group 1), but you didn't mention specifying this call group in the Call Group field of these extensions.
Are you possibly missing the configuration of call group(s) for your extensions?
To achieve a configuration that would put all extensions into the same call group and allow them all use Call Pickup for any other extension in the same group, you'd have to put "1" into both Call Group and Pickup Group(s) fields for all extensions.