How to add an article to existing replication
Adding an a new article involves adding the article to the publication, creating a new snapshot, and synchronizing subscriptions to apply the schema and data for the newly added articles.  Reinitialization is not required, but a new snapshot is.

Your steps are correct if the publication only has existing push subscriptions.  If the publication has existing pull subscriptions, you will also need to execute sp_refreshsubscriptions.  And you are correct, to generate a new snapshot you can use the View Snapshot Agent dialog, or you can execute sp_startpublication_snapshot.

To avoid generating a full snapshot when adding a new article, publication properties @immediate_sync and @allow_anonymous must be set to 0. Execute sp_addarticle, then execute sp_addsubscription. If the publication has existing pull subscriptions, you must also call sp_refreshsubscriptions. Then generate a snapshot and only a snapshot for the newly added article will be generated.  Note that this approach is not necessary but is typically used when Administrators want to avoid generating a full snapshot when adding a new article.

Please see Add Articles to and Drop Articles from Existing Publications for more information.