Nah not soft start -- that'll take too long, and have other drawbacks (e.g. a 100uF+ SS cap being discharged through ESD diodes when powered off, maybe who knows?). You'd have to somehow match the SS and supercap (initial) voltages and values, or take much longer for it to reach that point, it's weird?
Just current limit, so it delivers constant current into the load, and the load does whatever it does (i.e., I/C = dV/dt).
The downside is if it ever ends up shorted, it's just sitting there cooking forever, drawing supply current and dissipating some power (not necessarily problematic, just that it's more than zero). Maybe a foldback or hiccup limit, just with a lower threshold voltage, or a much longer timeout before faulting occurs, would be most attractive here? But again, keep in mind, this is already looking beyond a single failure event, and needless to say, the backup cap isn't backing up anything if it (or something connected to it) has failed shorted, and a lot of designs just aren't concerned with how they fail beyond a single event.
And yeah actually, it might need to be a low dropout buck. This is usually specified as having a high maximum duty cycle, >95% say, or 100% specifically. Then the output can be steady on, and the regulator acts like a low-drop switch between VIN and VCAP. It could also be a regular type, supplied from a higher rail, with the Vout feedback set appropriately (say, 5V supply charging a two-cell cap up to 3.3V for 3.3V logic use).
The LDO comments specifically are in regards to a pure analog solution, using a Low Drop-Out (LDO) regulator instead of switching. More power is dissipated, but only while charging, and then it can act as a low-drop switch from VIN to VCAP.
You wouldn't usually use "LDO" to describe a switching converter. But in the same sense that that applies (the maximum duty cycle), yes.
Tim