In short, PublicInput does not integrate with Webex Events or WebEx Webinar. You will need to create a basic WebEx meeting to integrate the PublicInput phone bridge.
An easy way to tell whether a meeting is a WebEx Event or Webinar meeting is that it will have 2 separate URLs, one for panelists and one for participants, where the latter will join the meeting muted.
Since there are 2 meeting links, the phone bridge cannot join the call, which means you won't be able to allow participants to participate by phone or listen to the meeting. You will also not have a transcript of your meeting and Closed Captioning will not work.
There is no need to have separate meeting rooms (panelist/participant) with PublicInput; you are only inviting the meeting presenters to your meeting, with the participants joining via the meeting phone number (audio) or on the PublicInput meeting page, where they can view the meeting stream (if streaming) and interact by chat/phone, etc.
See also:
Can I use any version of Zoom to stream my meeting?