I'll point out that you did not answer the question I posed so I'll ask again: Why does it matter that the player is choosing the action to "achieve a game effect?"
Every chosen action in an RPG is to achieve a game effect.
You have said repeatedly that suiciding your frog-PC is a "reasonable" and "valid" action. I contend that it is not without in game justification for the action. Therefore it matters why the player is choosing that action at my table. I'm fine with it, if they have a basis for the action in-game or in-character. Because I know the PC's at my table, I know already for which one's that action might be reasonable, so when I ask for an Arcana check, the DC is set accordingly.
The question under discussion is not whether table rules exist or should be followed once agreed to, but rather why some people take issue with how other people choose the actions they do, if those actions are otherwise reasonable in context.
I continue to maintain that self-suicide is not a reasonable action without an in-game basis, regardless of how you try to achieve the effect, because the out of character rationale is to end the effect of polymorph.
As has been shown, knowledge of the spell is not required for the toad to take the action. If you do declare that it is required, then to be consistent, no toad in your campaign world can ever accidentally die unless it has knowledge of spells. Is that true of toads in your setting? I imagine that is not the case, but then we are talking about storytelling in worlds of swords and sorcery that shares elements with childhood games of make-believe so anything's possible, right?
Knowledge of the spell IS required for a PC to take an action that would end the effect of a spell in my games though.
It's not an accident if you purposefully do it and I'm not sure how or why you continue to conflate them. The toad can't do anything on it's own to die other than jumping into an environmental hazard. That wouldn't be accidental unless as the DM, you said the player had to roll a d12 to determine which "random" direction their frog was trying to go. Otherwise it's purposeful and not accidental.
On what basis are you making this assertion? Can you reference anything specific to back up your claim? I'm running an Eberron game right now so I'd like to know for my own purposes how common 4th-level spells are.
5e based reference, but there are more from 3.5 forward that deal with the same information:
http://keith-baker.com/common-magic/
Two 3.5 based articles that deal with the rarity of PC classes and levels which then also speaks to the rarity of higher level magics/abilities:
http://archive.wizards.com/default.asp?x=dnd/ebug/20041227a
http://archive.wizards.com/default.asp?x=dnd/ebds/20040712a
Does your claim apply to homebrew worlds as well?
Obviously not and now you're nitpicking. When I said "established game worlds" I was referring to "official TSR/WotC published game worlds that I have read, or seen, or played in". I am in no way able to speak to people's homebrewed worlds. I'm sure someone or many someone's out there have worlds where janitors have access to 5th level magic on a regular basis.
Still, that would be an in-game justification for the character knowing how to end the polymorph effect. I'd still require an Arcana check to see if they had experience with it to act accordingly, EVEN in that kind of homebrew world. But the DC would be lower in that kind of a world.