Some situations warrant those rolls however. The tadpole and Nettie examples are perfectly fine.
-The tadpole example is the tadpole in your head working against you. Its trying to survive and keep one of its companions alive.
-Nettie is also a very good example of well placed dice rolls. You basicly told a druid that you have a parasite in your head that will change you into a brain eating alien. She doesent have the expertise to heal you and the one person who does, is missing and might be dead. To protect you and others she goes for the most humane option available to her. She is basicly giving you a mercy kill to make your passing as painless as possible while also keeping the people around you safe. Convincing someone in that situation to change their mind SHOULD be insanely difficult.
I think Larian did a very good job here personally

As a DM I'd never do this as it wastes time, and that is echoes in others' responses.
If there were multiple fail states for the different checks applied, happen over the course of a huge chunk of the adventuring day, like with downtime activities and the differing rewards from such; one could argue this situation would warrant 3 checks.
That is not the case here.
Passing the first checks mean nothing as the dialogue progresses anyway toward the final one. Failing the important ones thereafter is a pass or fail situation that just pointlessly adds a another layer to such. And then the final check (to lie about whether or not you'll take the wyvern Venom if things get real bad) is essentially a conversation ender with strings attached.
This is the furtherest from how any DM worth their salt would handle this situation, not even in games made pre-5e. There's always thay old addage "Don't make the party roll on things that's already going to happen, as you'll just make them feel like their choices don't matter."
Edit: a solution to this would be to have these options presented through dialogue, like normal. And/or provide them based on Nettie's disposition, which would change with the dialogue chosen through the conversation and affect the check at the end.
What you say should matter.