Do softer tires provide better suspension?
Most definitely.  I have some Schwalbe Marathon Plus HS 440 - very puncture resistant tire.  It is hard rubber and has a puncture insert.  It is a hard tire and gives a hard ride anywhere in the pressure range.  I have a softer same size less less ballistic tire on another bike and at the same pressure it is a softer ride a nicer grip.  And it is not the bike - I noticed the hard ride as soon as I put the tires on the bike.

Is it a mountain bike?  You want a soft ride then put some 2.2+ tubeless on it and run at 30 - 40 PSI.