The official script for the episode has been released and shares some interesting insight into the trial. The main thing being that everyone in the room apparently already knew the trial was for Littlefinger, not Arya.
SANSA: How do you answer these charges, Lord Baelish?
Littlefinger stiffens, as everyone in the room looks at him.
Everyone else is in on it. For once, Littlefinger is the man on the outside.
As if caught in a bad dream, he cannot speak for a moment
Game of Thrones, Season 7 Episode 7, "The Dragon and the Wolf"
By doing it in trial they expose Littlefinger for who he was to everyone, not just those who already knew it. Though to be fair most people already knew and those that didn't were told before the trial as per the official script.
What can they gain from doing it in secret?
- Vale? They already have its support with or without Littlefinger due to being cousins. Lord Royce seems to support the Starks and with Littlefinger gone he will most likely become 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.