A webhook / connector is one option, but it's limited to team channels, as you've noted. To send messages in a private (1-1) or group chat, you can use either a bot or the Graph. It depends if you want to send to others, but from an application (like is it ok for the users to get a message from a "bot"?) or if you want to send as if it came from you.
for option 1 (from a bot) you will want to look at "Pro-active messaging" (I've also got some answers here on Stack Overflow that you can search for, under my profile).
for option 2 ("as if" the messages came from you), use the Microsoft Graph, something like this: https://learn.microsoft.com/en-us/graph/api/chat-post-messages?view=graph-rest-beta&tabs=http .
