No, using would not fit.
This is the noun use (pronounced with an /s/ sound, not a /z/). That has one meaning which is something like the act of using, but here it has another meaning which is approximately usefulness.
In fact, of use is an idiom which means the same as useful.
So if you saw
Use of this equipment is prohibited.
that would be the first meaning, and so Using this equipment would be equally good.
But in phrases such as of use, any use, no use, it always has the second meaning: you could substitute usefulness, but not using.