It is quite clearly stated in both this announcement and the docs [1] that it is available for the personal (aka free) plan. Not sure why their pricing plan does not include any mention of it.
What you are describing is quite literally the UX, the user experience, and Slack overall has quite a good one. Although several of the newer features are somewhat lacking in that regard, like the threads feature.
[1] https://tailscale.com/kb/1193/tailscale-ssh