Yeah, that circuit is roughly the right idea.
A current mode hysteretic control is a good and cheap way to implement this; consider for example the classic L297/L298 stepper driver system. These sense ground-return current and PWM the load (which is basically a solenoid) to control current. When turning on or off, the load gets a large voltage (i.e., full supply), for high [current] slew rates. When idling in a state, the load's average voltage is chopped down.
Those two chips are... terribly dated, but there are newer options, or you can make your own.
I think there are solenoid controller ICs out there that can do everything in a few parts, but a discrete version could be done in, hmm, a dozen or so parts, if carefully optimized and you don't mind a few missing features (like wide supply range, super high efficiency, and various kinds of protections).
The power stage isn't very interesting; the one- or two-switch version (e.g., compare the EDN example to the injector example above) can be used, with the downside that the one-switch version has to dump the excess coil energy (which, it can be dumped into an auxiliary power rail and converted back into the supply to avoid burning it), while the other circulates the energy by itself (a pretty clear winner, unless some kind of economy-of-scale should apply).
The two-switch does have the downside that you can't use a common-ground (or +V) load.
The EDN circuit I think uses a design intent from a handy philosophy: keep everything consistent if possible. Namely, all you really need is a current mode converter for the power stage, and then you can vary the current setpoint to get the leading step. It's always in control, making your circuit immune to destruction due to a shorted load. You don't need a bunch of arbitrary junk -- state machine and timers -- to run it. (Granted, in this case, the state machine approach may well be simpler.)
Other gotchas include switching waveforms on a wiring harness, and general automotive concerns (e.g., reversed battery, load dump, smoking high temps in the engine bay). You would at least want those cables to be shielded up to the injector/solenoid, or to put the circuit right on top of the device. The former adds expense, the latter adds complexity (high operating temp).
Tim