I think the high-level idea is super cool! I feel like it will really help me be more productive. Dividing time into projects just makes so much sense. And the time bar for entering the number of hours is super smooth interaction. The idea of the game is good. But I have used an app that let me plant trees by not touching my phones for a few hours and that actually become more annoying than anything. If you want to make it good, you just have to go to higher fidelity. Because the success of the design I think has a lot to do with the passing of time.

A few comments on the lower level interactions:

1. Add task and hours together is a bit strange. When you click on the card, the cursor is at the “task” section for text, and there is a bar at the bottom. My first reaction is to type “3” hours because a lot of other tools do that. I understand you want to simplify, but probably more successful to define what is entered first and then next, so people do not have the choice to do either time > task or task > time and get confused.
2. Entering the time like 9:30 or changing the hours from 3 to 5 hours. It is not that big of a deal but you might just need a tiny onboarding. Otherwise I can see people get confused about how to do that because it is not obviously that they are clickable.
3. For sharing the timesheet, first I think you should have a “select all” option. Second it will be nice you can divide people into groups. I imagine Project X might be for client A, and project XX can be for group B etc. Maybe there will be a section for “frequent” too.

But overall great job! Hope my comments can make your project even better!