Why didn't Varys just kill Tommen?

[Usman Mir]

You just answered your question yourself.

Killing Tommen would've made Myrcella the last remaining heir of the Royal Family and Dorne in power as Myrcells is with them, however what Varys wanted was to make the IT weak, not heirless and open for anyone; so that when Aegon arrives at KL he can win the throne by defeating the one sitting on it or possesses it but that would not be possible if the last remaining claimant is not in KL but in Dorne.

On the other hand, Varys wanted Cersie to be Regent again and start ruling again which was only possible if Tommen is alive and can be used as a puppet like she did in AFFC.

Also with Tommen dead and Myrcella in Dorne, KL will have Tyrells in power and they have the power to oppose Aegon as well as Stannis making the IT again strong. Killing Tommen would've raised another Clash of Kings but keeping him alive makes it weak and easy to conquer.

[IRON BANK]

To be honest I feel keeping Tommen alive with Cersie at the helm is more destructive than killing Tommen. If you kill Tommen the chaos unfolds before Varys is ready to act.

[Iona]

Is this true? If so it kind of undermines the fAegon argument that Varys lied to Kevan about Aegon because the children were around to hear it.

It's strongly implied that the little birds don't get to keep their tongues in the Varys/Illyrio conversation Arya overhears in AGOT. There are little birds around when Kevan gets killed, and it's very queer that none of them say a word, not even the one that came to fetch Kevan.

ETA: the little birds are literate, so they do have a way to reveal secrets, otherwise they'd be pretty useless to Varys.

[Carter209]

Varys needs Cersei in power to completely destroy any hope of salvaging the Southern Alliance. He'll probably use his puppet in KL(the High Septon) to execute Marge, while UnGregor acquits Cersei in a Trial by Combat. Think of it like this: who, PR wise, should Aegon fight: the former(and some, current) Targaryen-loyalist Tyrells, or the architects of the Sack of KL, the Lannisters? They have no competent commanders, their fighting forces are arguably the most spent of any region, and their home region is the furthest from the capitol leaving the Tyrells as its only viable defenders, and they have their own issues with the Ironborn.

[Alta1r]

ETA: the little birds are literate, so they do have a way to reveal secrets, otherwise they'd be pretty useless to Varys.

Ahh, I see. I didn't realize they were literate. Although now that I think about it, it wouldn't make any sense for them to be mute and illiterate, lol. Doh!