Archer and Tucker weren't watching a broadcast game, they were watching a recording sent specifically for Archer. Hardly the same thing. Besides, since 99.99% of the time in Trek the characters are nowhere near Earth, they wouldn't have access to broadcast programming anyway, would they? Also, consider that the 22nd, 23rd, and 24th centuries are all very different, so it's not reasonable to assume that what exists in one would exist in all. For instance, films were still produced in the 22nd century, with awards going to the historical WWIII epics, but the only acting we saw in the 23rd and 24th were stage plays.