I am asking the second question you posed. "Or are you asking which one is more important from a "make it a habit"/track the time spent doing it, perspective?"
I am trying to decide which behavior is best to monitor.
Well, I won't cop out and say "track both". As stated, I'd track the weekly review as it's the more strategically important habit. If your weekly reviews are spent largely on just processing then that is an indication that you need to spend more time each day getting your inbox down to zero. Not helpful I am sure but as you said, it's a false dilemma.
However, a common problem that folks have is that during either the weekly review or processing their inbox, they start doing the work which makes the processing take much longer. That could be the root of the issue. Other problems include things such as having a system/tool that you just don't like using/interacting with or needing to reset expectations with people providing inputs into your system (e.g. letting people know that they may be waiting for 24 - 48 hours for a reply to their email) causing you to get overwhelmed.
I don't want to go too off topic and turn this thread into a solution'ing session, but I suspect the root issue(s) here is not which activity you track the time spent on it but something else entirely.
I hope this helps