By doing it in trial they expose Littlefinger for who he was to everyone, not just those who already knew it.
What can they gain from doing it in secret?
- Vale? Already have it's support with or without Littlefinger due to being cousins. Lord Royce seems to support the Starks and with Littlefinger gone he will be Lord protector until Robin comes of age.
- Littlefinger's contacts? How will they know now? After someone dies you don't magically learn everything about them.
- Influence? They are already the ruling family in the north and essentially the Vale, what other influence do they need?
What can they lose from doing it in secret?
- Trust
- Power
- Support
- etc.
Another good point raised by @Odin1806 is:
In addition they also show that the Starks are a unified front once again. While some may have believed (and rightly so) that the house Stark was in shambles holding a trial and making everything public shows that the Starks are all home, they are back on top, and they are not to be trifled with.
As to why Arya didn't take Lord Baelish's face, well we don't know if she did or not. For all we know her and Sansa are having a lot of fun with it.
