Yes, even with all of those injuries and late-in-their-career veterans who have manned first base for the Nationals, Larry Broadway hasn't been one of them.
There is no reason to believe that Larry Broadway would develop into an all-star caliber player, but it's not out of the realm of possibility that he could give the Nationals at least what they've gotten out of their fill-ins this season. Just this season, Ronnie Belliard, Aaron Boone, Kory Casto and Paul LoDuca have filled in at first base and those four have a combined mark of 16-78-.242 and that includes the respectable numbers for Belliard, who has hit 10 home runs and is hitting .271 on the season. Broadway would also give the Nationals better defense at first than any of those others have.
One reason why Broadway doesn't get a shot at the majors is that he's limited to playing just first base. While he's very strong defensively at first, he can't legitimately be considered ready to play other positions although he has played very sparingly in both left and right field this season. In fact, he's also pitched in five games for Columbus and has a 2.70 ERA in 6.2 innings of work. Maybe that will be his key to reaching the majors.
The bottom line: Since Broadway is not on the 40 man roster, his chances of getting a shot this September aren't very good. He'll be a minor league free agent after the season and it will be interesting to see if he winds up going elsewhere to look for his chance to reach the majors. He's not a stellar player and is getting a little old to be considered much of a prospect, but it would be interesting to see what Broadway could do with a shot at some playing time with a major league club that is going nowhere and doesn't have anything to lose; oh, wait, that describes the Nationals.