I'm thinking on if the Empire has any wardroid around. Marvel and Dark Horse did make mention of such things being used from time to time to supplement the stormtroopers and walkers. Not in large numbers, but usually a small but powerfully armed wardroid. The Empire not using large numbers of combat droids because of the fallout of the Clone Wars. The fear of the Seperatist's droid armies, and the huge number of surviving Clone Trooper (now Stormtroopers) followed by the training of new troopers for the armor, rather than build a droid army. But troopers sometimes could use a droid as backup fire support.
The thing about droids as weapons of war is that individually, they're not very bright (by design, since you can't trust bright murderbots not to get ideas about who's giving the orders) and their only real advantages is their large numbers and that are relatively cheap to replace. Their main weakness as displayed in both TPM and RotS is that they can all be shut down en mass at the push of a button (again, by design, since it's a bad idea to have a hundred million murderbots unless you have a way to make them all stop murdering in a hurry.)
For those and other reasons, I doubt the Empire used battledroids in any major way. Specialist assassin droids, espionage droids, perhaps the odd sentry droid and of course armed recon droids like the probot are probably about the extent if it.
Stormtroopers by dint of being living thinking beings are inherently much more flexible, imaginative and better able to work in teams and react quickly to changing circumstances are one-to-one, the superior fighters.
Now using conscripts from backwater worlds also has it's drawbacks vs. legions on clones designed and trained from birth to be born soldiers also has it's weaknesses (the inability to shoot straight apparently being the most notable) but I suppose one might argue that draining as many young, idealistic people away from a civilian life and into the highly regimented world of the Imperial military gives Palpatine one more lever by which to manipulate the galaxy. Plus there may very well be a force related reason for having billions of living beings dedicated to (and presumably terrified by) the only two Sith in the galaxy.
There can't be much attraction to a dark side user to rule over an Empire made up of lifeless machine. Much better to take the life that is already there and get it to think and behave like one big machine instead (figuratively speaking.)
