I'm pretty sure the answer to this is "Yes" but if I buy the commerce add-on, can I use it to create an "Ad free" membership that automatically adds users to a group, then use that group to hide adverts for those users?

In a word, "Yes".

You can configure Commerce to either change a Primary Group, or add a Secondary Group, for a member.

The Group you assign will determine what they can see, based on what Group permissions you have setup, and you can set Ads to show only to selected Groups.

Please check our documentation for Commerce, starting here: