Can someone of higher rank chose to go outside the wire?

SGT (Join to see) Four Star. GEN Petraeus routinely went "outside the wire". My Division Commander did during OEF. Brigade Commander was out all the time. All the Battalion Commanders in the last BCT I was in, ALL kitted up and went on ops. They command their unit and d what they need to in order to accomplish that.

It's called Battlefield Circulation. Leaders do this all the time. They go where ever they think they need to go to lead their unit from the decisive point. It is routine for BCT and BN commanders to have their Tank, Bradley, Stryker to fight off of. Absolutely common to have Battalion Commanders out on ops. My BN Commander went on patrols frequently. I went as a SPO but not as an XO, because my job was to be there to support my BC going where he needed to go, patrol, airdrop, other FOBs/units.

Your PSG can decide to go outside the wire if there's a valid reason. Going outside doesn't mean they will see combat. Out of two years in Iraq, and several hundred combat patrols, I've only been shot at a handful of times, blown up even less, and had even fewer enemy combatant situations. Although, I've had at least a few hundred mortars and rockets shot at me.
Point is, by the time your BDE CDR gets there the area is pretty much settled. The majority of the combat patrols your platoon undertakes are boring and uneventful. If a BDE CDR was looking to get into a firefight that day, they would need to be in a hostile area and go visit their most remote outpost. Even then, the majority of the patrols are going to be nothing. Colonels have more important work than walking around third world villages hoping for someone to shoot at them. When you have 5,000 Soldiers you're worrying about, you just don't have time for that. No leader has time to mess around with that kind of thing, and if they did they would probably be a narcissist

Having never been afforded the opportunity to serve as a senior officer... I never the less often did consider what, "I would do differently", had I been. I think it's safe to say every good officer wants to share as much of their peoples' burdens and risk as possible and practical. As the S-3, I routinely put myself on every convoy I planned (usually in the turret)... until a senior enlisted member counseled me to remember that any "good intentions" I had, may have had very "bad" unintended consequences. There are times when a more senior person's presence contributes... and times when it unnecessarily complicates things and/or builds resentment rater than respect. I once heard it stated like this; we all have our "picture" that we manage. In a motion picture, the actors play their parts... and the director manages. It's hard to envision Spielberg trying to manage "Saving Private Ryan" as an extra on set. That said, there are plenty of examples from history where a senior officer's direct involvement on the battlefield made a very real difference... and some not so far back in history (Jim Gavin, Patton, and Mattis are just a few examples). Myself personally? I would be highly motivated by seeing a flag level officer "out front"... unless his being there was in any way detrimental to the safety of others, or the success of the mission.