So I think that the current structure of hardcore coop really fails to make the whole thing feel more challenging or even curb the rampant rushing. All it dies is make you feel handicapped and the whole exoerice even more of a mess than usual.
The reason IMO is simple; there's no actual pressure to not die. The rusher will go out and respawn the next turn, swap his mosin for a dropped gun, rinse and repeat. The mosin makes dying feel like a penance, but doesn't really make you feel pressured to try to stay alive. The structure may work when you know who you're working with, but falls apart when applied to random people on the internet who may or may not communicate, since it lacks any actual fair feeling incentives, and the handicaps fail to actually make you feel like you need to stay alive in the long term.
Now one solution that I can think of is to make respawns less frequent and add in more objectives while keeping everything else the same as standard coop (better AI, fewer enemies, but no more mosin or lead weights on feet).
The basic idea is that resupply crates and respawns only appear on the capture objectives; no resupply or respawn on destruction objectives. However, each capture objective would be interspersed with more destruction objectives; so instead of blow up one cache and then cap a point, it would be more like blow up 2-3 caches before capping the point.
Now, of course this would require some work in exactly where the various objectives would be placed and what types of objectives go where, not to mention a few alterations to bot loadouts to remove molotov and grenades (in order to prevent them from scavenged and used to quickly destroy objectives, not to mention reduce the arbitrary deaths from laser guided grenades); but I can see a number of benefits to this.
This system would allow for hardcore coop to feel more pressured and place a greater incentive on staying alive (since you respawn less often) and judicious use of deplorables (since you resupply less). It would also curb rushing without any need for handicaps (which don't really seem to work anyways) since you wouldn't have any sort of assurance of just coming back into the fight, and even the stuff that the rusher does bring wouldn't be assured to be enough till the next resupply (thus promoting its judicious use). Best of all, it would do so without actually handicapping the player while still making them feel much more challenged.