One reason to stick to a "deposit the check, write a check for cash" rule is so that your tellers have a single, simple rule to follow. They don't need to concern themselves about the status of a business as a sole proprietorship.
I completely agree with this. This has been the policy at most of the FIs I've worked for, for precisely that reason. We all know that sometimes the "status" of a business is not very clear. It is much cleaner to have a policy to deposit all checks payable to a business entity.
For what it's worth, one of the FIs I worked for used to allow for the checks to be cashed (as well as give cash-back on deposits) and ended up being involved in a lawsuit because of some improperly cashed checks. They changed their policy very quickly - business deposit tickets don't even have a "cash back" field anymore.