smudge effect on the edges

I would caution your customer on the fact that as the artwork gets sized down, the effect becomes somewhat indiscernable.  I did notice a grunge effect in your blur version, sort of a natural brush effect.  But, I wonder what purpose a soft edge would serve?  Logos, in general, are simple.  Further along in their application, they can be modified in a particular use, such as a poster or video.  Why would your client want to make their logo more complex now rather than later?  To answer your initial question, Illustrator can be used and I am thinking of a Blend, but would not recommend it.  There are many grunge type fonts available, as well as your handlettered rendition.  I also wonder why that would not be enough?

This can somehow be done by creating a brush that can make this stuff and then drawing additional lines and apply that brush.