Softness per se is no problem.
It depends a bit on what you see as a problem - a soft suspension means the equivalent volume of air (Vas) will be very large. This is even more true if the piston area (cone area) is large - a large piston compresses more air for the same piston excursion, so to keep the air-spring equally soft, you need an even bigger enclosure.
So that means the "light cone, soft suspension, large cone area, low resonance frequency" driver is likely to need an unacceptably large enclosure volume by todays standards.
In this era of gobs of cheap, clean, audio power, it's a much more viable option to make a heavy (and yes, less efficient) speaker, because it will allow a smaller-volume enclosure to be used with it.
A couple of the 15" woofers I looked at on the Parts Express website had sensitivities around 91 dB per watt at 1 metre, so efficiency is not too shabby, heavy cones and all.
One would think the smaller Vas is less important for dance-club installations, but I think even there, there is now less tolerance for having to deal with 50-kilo speaker enclosures the size of small refrigerators.
I remember a Carver subwoofer from the late 1990s that had a 10" woofer, and a 10" passive radiator, mounted on opposite sides of a cubical enclosure that was only 11"x11"x11" - less than one cubic foot
on the outside, not to mention it contained a power amplifier inside taking up even more air volume.
Both the woofer and passive radiator were almost ridiculously heavy, resulting in a pretty low resonance frequency - claimed to be 18 Hz - as installed in the tiny box.
Carver's trademark too-much-power-is-just-enough approach completed the design, with 2700 watts (!!) of built-in audio power, if you believe the manufacturer's claims.
This "irresistible force meets immovable object" design supposedly resulted in deep, loud, bass from an almost absurdly small speaker system.
I found a little more information on this bizarre beast online, here:
Welcome Secrets of Home Theater and High Fidelity
-Gnobuddy