Even if you use the 'diffuser' part of the speedlight?
Yes, that can work.
The problem is, to fill a softbox evenly with no hot-spots (or darker corners) the light has to radiate all around, get bounced off the sides and nicely scrambled. The circular flash tubes in studio heads do this naturally and softboxes are usually designed with studio heads in mind. To make double-sure, they usually have a second diffuser panel positioned inside.
With a speedlite, all the light goes straight out of the front, hardly touches the sides, and hits the diffuser front pretty much directly with little diffusion. So the centre is brighter than the edges unless you find a way to spread it around more. The diffuser wide-angle panel can help a lot, sometimes offering a pretty good solution.
For example, if the wide-panel will cover a 14mm lens (on full-frame) that's 104 degrees side to side, but only 81 degrees top to bottom. If the softbox sides are at maybe a 90 degrees angle, that will spread light to the sides quite well, but not top/bottom. That could be enough, and the way to test it is to shoot the softbox itself, square on from the front, and adjust exposure so it's just around the blinkies threshold. This will show the hot spots very clearly.
Another trick is to fit a Stofen diffuser cap (or similar, only a few quid off Amazon). Test it as above, and to get more light out around the sides, cut/drill some holes. It's easy enough to get very good results this way, but needs a bit of DIY and trial and error. Start with just a few holes, maybe 1/4in, then a few more until it's right.