There is a step 2.2 after step 2 before step 3 that I found helpful to remove a Gmail SMTP server setting that should've been removed when removing the Google account, but sometimes is not.
After having removed the Google account, but before setting it up again, open Mail right now. Go to Preferences > Accounts. For each account in the left pane, make note which "Outgoing Mail Server (SMTP)" is shown in the "Account Information". Typically, this is pretty straight forward for most users, but can get tricky, if you get mail for multiple roles.
After taking note, select any remaining Account. In the "Outgoing Mail Server (SMTP)" box, select "Edit SMTP Server list". In the dialog the opens, check that there is no server smtp.google.com in the list. If there is one (or more), select the server, and select the minus "-" button to remove.
After all Google SMTP servers have been removed this way, close the "Edit SMTP server list" dialog that shows the remaining server by using Cancel. After all, you don't want to change the SMTP server on the 'any' account you previously selected. Now, in the left pane showing all your accounts, click on each account on the left pane, and verify that the "Outgoing Mail Server (SMTP)" in "Account Information" is still as before. If it is not, select from the drop-down the proper server, (use your notes from above, if in doubt), and then click the next account. That will trigger the dialog asking you to save the changes, which you do. (At the end of the accounts list, select the first account to trigger the dialog.) The "save" dialog will not appear, if you didn't need to make any changes. But do check every account.
Now, quit Mail, and proceed to step 3.