Spring frame in place of shocks?
Softride is one of the bigger names in this area.  Their main bike design exchanged the top/seat tube and seat stays for a composite materials &quot;beam&quot; that cantilevered back to a saddle in the regular position.  Originally for MTBs, they pivoted to Triathlon bikes in the mid 90s.  The company stopped making frames in 2007 mostly because the design was no-longer UCI legal.  The frames never conformed to the double diamond required of road bikes, but tri bikes had more lenient rules.  While softride still exists, they make tailgate protectors and straps.
By AndrewDressel - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=62716747
Notice the chainstays are much more substantial and therefore heavier, to hold the rear wheel.  The stub top tube is also beefy because not-only is it holding much of the rider's weight, it is levering downward also requiring strength in the fastener.    As such there are much smaller weight savings than you might imagine.
Another downside is that the suspension beam is made of slightly flexible material (actual composition is unknown) but imagine that snapping and dropping the rider down.   The beam should be engineered to minimise sideways flex and twisting forces, I don't know how well they succeeded.
More info  https://en.wikipedia.org/wiki/Softride

Your idea of a flexible frame will have poor handling characteristics in a corner.  If the effective wheelbase can change, then the bike will get longer and shorter which will make cornering feel weird.
While its a known technique to brake hard on the front going into a corner, to load up front-fork suspension which effectively shortens the bike for the short duration of the turn, having the frame change from long to short to long with every pedal stroke will be disconcerting to the rider.