Hi Trevor,
You've got quite a few elements in your description that are quite specific to your business, the only specific business use case the Microsoft support is more of an appointment booking solution through Microsoft Bookings. I think to meet your requirements I would consider using the PowerApps service provided as part of Office 365 to build something that meets your needs exactly. PowerApps lets people build their own user experiences on top of the Office 365 services without extensive development experience, most people who could build macros in Excel could build an application in PowerApps. PowerApps can be added as tabs in Microsoft Teams, and you could develop your app to post information into a Teams conversations as you required. Behind the scenes you would probably store your list of clients, teams and then booking in SharePoint lists within your team. It sounds like a fun project for about a week of development.
If development was really out of the question, then using Planner through Teams would get you some of the way. With Planner you could create a task for each of your jobs and assign Team members to each task. Maybe using tags to indicate equipment etc. There's a calendar view so you can see when tasks start and complete etc.
HTH