Free week serves multiples purposes. If it only served the purpose of "try before you buy", then it would make sense to remove the 450 IP champs from it. But because Riot wants to include at least some of the easier to play champs for newer players, that's not going to happen.
Part of the reason Ashe shows up so often in free week is that the pool of ADC champs is so much smaller than mid, top and jungle. The pool of support champs is also very small. The pool of champs for Mid, Top and Jungle are at least 50% larger than the pool of ADCs and Supports. Because of that, I'd like to see a 50% increase in the number of Mid, Top and Jungle champs in the free rotations. In other words, add 1 more of each, for a total of 13 champs on free week.
That mitigates the problem to some degree. Though it still doesn't entirely deal with the fact that players are forced into trying out champs on Riots schedule, rather than their own. In general, I'm not a fan of when games dictate when I should play a certain way. For example, if I see a champ on free week that doesn't show up as a free champ all that often, I feel like I have to try playing them this week even if I don't really feel like it. If I don't, I may not get another chance for quite a few months.
It's something I've tolerated so far, especially considering it's the only way that LoL forces me into playing the game in a way I might not otherwise play. Heroes of the Storm, which has been pointed to in this thread as a model of good design, pushes you into playing certain champs on certain days. Want to do daily quests for gold to unlock Heroes? You might be forced into playing a hero type that you don't enjoy, or from a group of heroes that you don't like. With LoL, the equivalent is the first win bonus, which is far more flexible.