Best approach to extracting a vehicle from a roadside ditch?

I had almost the same thing happen with my F-150 a few years back. Rear wheel dropped into a storm drain gap in a parking lot and the bumper was resting right on the asphalt lip. What worked for me was lifting before any pulling. I slid a bottle jack under the frame rail near the stuck corner, raised the tire clear of the hole, then filled the gap with offcut 2x6s stacked flat. Only once the wheel had something solid under it did I try backing out under its own power, slow and easy in reverse.

Pulling with a strap while the tire is still down in the hole is the part I'd avoid. All that force goes through one corner at a bad angle and that's when a control arm or tie rod gives up. If you don't carry a bottle jack, even the scissor jack from the trunk plus some flat lumber or stones can buy enought clearance to get the tire out and supported.

One thing I learned the hard way: look at the edge of the hole itself before you put weight anywhere near it. The concrete around my drain was old and a chunk broke off when I set a block down, which would have been much worse if a tire had been sitting on it. Take a minute to pick the solid spots for your jack and blocks, not just the closest ones.

I'd skip the lifting part entirely. My neighbor's minivan got a wheel wedged in a gap at the edge of his driveway apron, and my first instinct was the same as the original poster's, get the front raised. But the ground under the stuck corner was soft gravel, and the jack just punched down into it while the car barely moved an inch. What actualy worked was hooking a strap to the rear tow point and pulling backward with his truck at a walking pace, keeping the strap low so the pull stayed level instead of yanking the nose up. The wheel walked out in one slow, steady drag.

The way I see it, the bumper resting on the pavement edge is doing you a favor. It's acting as a pivot and holding things stable. Raise the front and now you're trusting the entire weight of the car to whatever blocks or cribbing you stacked under it, on ground you allready know is questionable. Pulling back the way it came in uses the contact you already have instead of fighting it.

One thing I'd add is to resist the urge to power out with the stuck wheel itself, that just grinds the tire and forces the suspension sideways. If it doesn't come free after a couple of gentle pulls, this is exactly where a flatbed with a winch costs way less than a bent control arm.

One trick that's saved me twice is letting some air out of the stuck tire before you do anything. Dropping the pressure gives it a squishier, bigger footprint and sometimes it just eases out of the gap on its own once the bumper stops hanging up on the edge. Reinflate after. Also worth asking, is the weight actually on the bumper or on the lower control arm? Makes a difference where you'd stick a jack, and some bumpers will just crack if they're taking the car's whole nose weight.

One thing I'd add is thinking about what happens after the wheel comes free, because everyone plans the lift and then forgets the exit. If the only clear path is forward, build a little ramp of flat lumber in front of the good wheel too, otherwise the car rolls off your blocking and the bumper smacks right back down on that edge. Also, before anyone touches a jack, I'd figure out exactly what's carrying the weight. If the fascia or a plastic air dam is what's resting on the concrete, it might crack or flex, and that changes how gently you want to raise things.

Not sold on the scissor jack idea from the F-150 story. Those little trunk jacks are made for changing a flat on level ground, not for lifting a car that's already teetering on one corner. The base is tiny and the whole thing can skid sideways the second the body shifts, especally on smooth concrete. I've seen one fold over mid-lift. If the car is resting on the bumper like that, the geometry is just wrong for a flimsy jack. I'd want a proper floor jack or a big chunk of timber cribbed up under something structural before raising anything, not whatever happens to be in the trunk.