Convolution integral for LTI system
How can the multiplication with the dirac function give x(t-t0) of every point in the convolution integral?

It can't, that is what the integral does.


  -- I try to imagine the convolution graphically.


This is very helpful, but inside the integral, it is unnecessary.
Inside the integral it is just multiplication.
The integral operation handles the infinite summation for all values
of tau.

So I agree, the multiplication cannot get every point, that is why the
integral is needed.

If you find a good video/gif of convolution
some here,
you can think of each frame as the multiplication for a particular
tau.  The combined effect is the integration.

Hope that helps connect the visual to the equation.