In your cleric example, the player is forcing the situation into the foreground. Backgrounding is more a "I won't ask; you won't tell" situation. Both sides agree that the a particular thing is just going to bump along in a satisfactory way without spending table time at it. If the player breaks the agreement then the GM needs to respond.
There is no loss of consistency from backgrounding any element just as there is no gain in consistency from forcibly including elements a game implicitly backgrounds -- such as daily ablutions and use of the toilet.
If you as a DM feel that you are just going to make the player jump through (almost) the same hoops every time the druid and T-Rex go to town but the town will relent and allow the obviously well-behaved, trained, and possibly magically controlled animal in, you might as well Background that hoop-jumping. The player asking for the Background is essentially asking "This thing is boring. I know what is required to move forward. You know I know that. Can we skip the time sink or do you want to watch me continually do this rather boring thing each and every time it comes up?"