You talked about hiding the geometry. Since geometric algebra is mostly about making it easier to visualise, why would you ever want to prevent that?
You would not, because that’s indeed a huge part of what geometric algebra really is. Whenever you try to solve an issue, you can always go back to its representation in space and have a solution ready for you. Because of that, even if something does go wrong, you can clearly see what you have right now and based on that search for the error.
The idea behind that was to build the bridge between this beautiful algebra of geometry and the techniques we are using (vectors, matrices etc.) to show how it all connects. It’s meant for people to find a way into geometric algebra based on what they already know.
Geometric algebra allows us to work in an endless number of dimensions, which means that adding time or space to whatever dimension we choose to work in is completely feasible. However, I can imagine this being very confusing to for example students in high school, who have only just started learning kinematics. All of the sudden they will have to work in an imaginary space. How do you see that?
It’s good to point out that, in the scenario you have given, an extra dimension is redundant. Whenever kinematics is implemented in a computer, it will work with the internal clock of the system. The algebra I covered is meant for physics problem solving. Think about space time paradoxes. These can be answered nicely with geometric algebra.