Strictly speaking it's not so much an issue on continuity as it is what makes sense for the character. Remember that being a Senator isn't a cover, she really is the representative for an entire sector of space with potentially hundreds of worlds counting on her being competent in the role. Spying for and aiding the rebellion is something she has to fit around that lest ISB mark her as a traitor. So like I said, it just doesn't make sense as an adventure show. It'd have to be some variation of political espionage thriller.
Well, I've just been rewatching The Clone Wars, and Senator Amidala sure got plenty of action-based episodes alongside the more political plotlines.
Also, how long has Leia been a senator at the time of ANH? She's 19, after all, so it can't have been very long. (US senators have to be at least 30.) But lots of people in times of war and crisis become effective fighters while still in their teens, and Jedi padawans certainly did in the pre-Empire days. So a series set before she took over as the senator for Alderaan could give her more leeway for action stuff as well as political intrigue.