How directional is LED strip lighting (with a diffuser)?

Even without a diffuser, the light should spread pretty well. With a diffuser, I would expect it to be fine.

For some time, I had a strip of LED tape used as the light for an under-stairs area. I powered it with a switched 8-battery AA box. The light was at least as even as any bare bulb would have been, probably more even.

(The battery-powered approach was a little dim, probably not good enough for a pantry, but it was easy and didn't require me to figure out any line-voltage wiring :) ).

I don't want to mount multiple strips pointing in different directions because that would probably be too bright

In my experience, "too bright" is not generally going to be a problem with the LED tape. I find the quality of light to be good, but never overwhelming. In any case, you can tailor the installation by choosing LED tape with the appropriate chip density for your needs. Densities of 60, 120, and 240 per meter are readily available.

There are also modest variations in brightness according to the wattage/chip type (e.g. 5050 vs 2835 vs 3528, etc.) but I've found that it's the density that is the most significant factor. The next most significant factor seems to be the color temperature, with higher temperatures being brighter.

That said, a single strip mounted over the door would probably provide sufficient lighting. Assuming you're mounting on or over the top of the door jamb, I recommend using a 45-degree channel for the mount, to aim the light down into the space.

Yes, an LED strip would work, properly size for the 4' width of the pantry and the brightness you want. The LED, located at the front edge of the pantry, should not create any more shadows than the single light bulb. In viewing your objective of a "Clean" look, This would take a lot of effort for the clean look in reviewing all your requirements. The light socket above would have to be converted to an outlet for the LED adapter to plug in, you would still have the issue of the wire running from the "outlet" to the beginning of the LED strip. So you would still see all the wiring for this setup. A "clean" look requires a lot of thought of do you want to pull wires on top of the pantry/ceiling or wiring channel around the edge. Also where would place the power brick, depending on the brand of LED-Power adapter you selected.

If the power adaped is not on the end of the plug, you could put the power brick in the corner, where at 8', it wouldn't be seen. so you only have to deal with seeing the wire from the outlet.

I just did the same thing a few months ago and it turned out great. I routed a 1/2" channel at the back of the shelves so the bulbs wouldn't be seen. The color of the walls is important as well. You don't want to reflect the light, but you don't want to necessarily absorb all of it. A few coats of a flat or eggshell saturated relatively dark color work well.