A fairly handy new app just released: #Aivi. It allows you to create iOS live activities that can be controlled via REST, for example via #HomeAssistant. Downsides: proprietary and another subscription, but at least it’s from some guy in Sweden, not a big company or shady startup. Will probably give the free trial a shot once I have the time, even though the pricing could use another look (2.99 € / month while still maintaining a limit of 5 activities is a bit much - I’d probably be willing to pay 12 € a year if it works well though, if that were an option).
More info: https://getaivi.app
Alternatively, If you want the HA devs to implement live activities directly into the conpanion app, you can vote for the feature here: https://github.com/orgs/home-assistant/discussions/84