It is possible, but I’m much less sure it could be done safely. The reason the password is randomly generated and emailed to the registrant is it confirms the registrant provided a valid email address that they have control of. This cannot be done unless a login is presented to confirm they received and read the email sent.
In order to go straight to the profile page, you would need to auto-login a new registrant, meaning anyone could register with bogus credentials, so there is really no point in registering at all.
You must present a login page to new registrants. It validates the email address and gets them to select their own password after the random one is used.
The default behavior after the login page is to offer the dashboard page. It is possible to change this to a different page if that helps any, but the intervening login page is really a necessity.
To go to the profile page instead of dashboard, you could hook into the ‘login_redirect’ filter and supply a different URL. This would apply to all users unless you use a conditional structure to apply it only to users of certain capabilities or lack of.
Or you could change the link in the sent email to that of the profile page. The login page will still be presented first. This is done by re-defining the pluggable function wp_new_user_notification()
. The original source code is in wp-includes/pluggable.php.