exactly, Relicator - some of the best and charming pubs/gastropubs in london are in Shep Bush and also Hammersmith on the river (easy reach).
Shep Bush is a large area - yes, a couple of roght estates, but if you aer central in london ,you can't avoid them. It's really not anti-social in the majoprity of the leafy and nice side streets, the only rough-ish area is high street late at night but no comparison to E&C or indeed Stockwell!
Shep bush spills out to Ravencourt park, as others mentioned and so did I - really nice, and a park with tennis court as i remember.
There are some little shop and more indep places to ear in Chiswick high road which is close, also Westfield and all that the other way if you are in Shep Bush proper.
Again High st Kensington, Holland park extremely close for more womdering around cafes, Op, but prices are so high - so imo Sh.Bush is still a very good value area on the fringes of these.
I think people confuse Sh.Bush with White city, to the north of westfield - that's really rough but it's self contained.
I'm VERY fussy about safety of areas, and even though it's not the safest of all, it's definitely overall safe, and I know london very well.
Agree it's densely populated but so it all central london. Belsize park less so of course but then it's less central. I'd recommend it too but ime ther is less to do there than o nthe doorstep of Shep Bush. Still an option of course.
OP you need to visit these areas and see for yourself! try to go by the exact addresses/streets taken from rightmove so that you don't wander into a rougher patch and then be put off!