The FBI is a police agency. It investigates criminal activity. The intent of an investigative board such as this one is to look more widely at the causes of, and response to the events on Jan 6th.
For example, it could investigate if Trump gave succour to the protestors, even if it was not criminal provocation. It could ask if the Police were prepared; if the policing response was appropriate; whether a more forceful response could have prevented the attack on the Captiol, or would have led to greater disorder. It could look at the role of the FBI and other agencies on the day.
All of this is beyond the remit of the FBI, who would not be an appropriate body to investigate the actions of the FBI, as that would be a conflict of interest.
A committee of the House could investigate this, but this is not really the role of legislators. Congresspeople are not investigators as their main skill. Congressional committees should be focused on investigations that are ultimately designed to guide the writing of new legislation. (This isn't always the case but in theory anyway).
Especially as one of the topics of an investigation would be actions that members of Congress. Setting up a committee of Representatives would lack the independence and legitimacy that an independent commission would have. Any House investigation could be hijacked by partisans, or dismissed as having been hijacked by partisans. The proper way to hold an investigation is to create an independent commission with the time to do a proper job, and not a committee of Representatives who are politicians and don't have the time or the skills to do this. However with the commission blocked in the Senate, one way forward for the Democrat-controlled House is to set up a Committee (that would have Democrat Chair and majority)