This may be more complicated and require customization (may require PS/CSM hours with your Customer Success Manager Evgeni Haikin) but here are some options...
1 - You have looked at Time Sheets, but have you tried using the Stopwatch? This allows you to have a running clock or stopwatch that you can manually start/stop when doing work. The catch is that it is based on a work item (i.e. Task). Here is a video that describes it...
2 - At the Customer Module, we are able to add related Tasks or from other side, we can add related Customer to a work item to link them together. This can be done in order to associate work with Time on it to a Customer.
3 - You could create custom Fields and Rules that are updated with hours and roll up as required. These would be created at the Customer Item Type so that the fields would be available within Customer Module.
4 - Custom Fields and Workflow rules would also be how you would approach your need to show data by Sector and/or Industry (Custom Fields).