I want to setup my bike to the most shock-absorbing configuration possible. Rear and front suspensions are set to "soft", will look for a shock absorbing saddle, but how about tires? My bike is a heavy e-MTB (Riese Muller), currently equipped with hybrid tyres. Keeping them soft makes a huge difference, at cost of a worst rolling(no problem). But how about the tyre type? Would a MTB tyre be softer or not, compared to a hybrid one?
The most important thing for shock-absorption in tires is low pressure. The next important thing is width. The wider the tire the more “suspension travel” you’ll have and you can lower the pressure even more without risking a “bottom out” when hitting a pothole or rock.
I think tire type or construction is less important, but generally high-end MTB (or cyclocross/gravel) tires are made to be run at a low pressure. This means a thin and supple construction (especially of the sidewalls) at the cost of being less puncture proof.
So first of all you should lower the pressure as far as possible. When I ride low pressure I usually “test” it by pushing the front wheel hard against a sharp edge (e.g. kerb). If I can make it bottom-out with my full body weight it’s just about right. If you ride carefully you might be able to go even lower, but at very low pressures there is danger of the tire “collapsing” sideways in turns.
Edit: I should add that MTB tires don’t necessarily have big knobby tread. There are also versions with almost no tread or very fine knobs available in case you mostly ride on tarmac or smooth gravel. They should also run smoother and be less noisy.