Absolutely. The position of company commander is generally held by a captain, so many captains would have been "at the point of friction" by landing with their companies (also there may have been some who weren't necessarily company commanders as well.) Remember that, traditionally and very generally speaking, Sergeants and above are responsible for their small teams, squads, and platoons while officers are generally tasked with understanding the broader picture and ensuring a unit's efforts are supporting the larger assault plan.

Amphibious assaults, like any well-planned military operations, are not simply a matter of dumping thousands of men into the surf line and saying "go!" Consider how complicated these landings were. There were multiple beaches, and multiples sectors for each beach, with specific units assigned to assault specific parts of the beachhead within each sector. In short, it was total chaos, and effective command and control was needed to ensure units remained as intact as possible and that units were assaulting their assigned targets (during D-Day, in particular, this part of the plan went to hell fairly quickly as ocean conditions caused many units to land far from their planned landing spots.)

Now, beyond this, also keep in mind that Saving Private Ryan fell somewhat into the Hollywood military movie cliche of having relatively high-ranking troops (in this case, a Captain,) slugging it out alongside the privates and sergeants. While this certainly did and does occur during combat, it was and is generally more the exception than the rule. So could a captain have single handedly led an assault on a German strongpoint? Sure, although in all likelihood it would have been an NCO or maybe a lieutenant.