I think the best example of Dany's first sign of madness is in Season 1, when she burned alive the witch, Mirri Maz Duur.
Think about how those scenes unfolded.
It's actually super deep to think about now that the series is finally over.
1. Dany finds Mirri Maz Durr (witch) when the Dothraki are killing and raping the people of her village.
2. Dany asks Mirri to heal Khal Drogo. Instead, Mirri uses blood magic to kill him because he's the leader of a group of murderers/rapists.
3. Dany demands that Mirri should be THANKING her because she saved her life.
Mirri responds with:
"
Saved me? Three of those riders had already raped me before you saved me, girl. I saw my god's house burn, there where I had healed men and women beyond counting. In the streets I saw piles of heads... the head of the baker who makes my bread, the head a young boy that I had cured of fever just three moons past. So... tell me again exactly what it was that you saved?" ―Mirri Maz Duur to Daenerys Targaryen
Dany then makes the decision to burn Mirri alive for killing Khal Drogo.
BDS mentioned in the OP:
"Dany tries to change the Dothraki and end their cruelty towards women."
But again: How does Dany bring about change to end the cruelty towards women? One of Dany's first major acts in Season 1 was burning alive a woman (Mirri) who was raped multiple times by the Dothraki. Before Mirri dies, Dany says she'll get enjoyment from hearing Miri scream as she burns.
Food for thought:
- Can we fault Mirri for wanting to kill the leader of a group that raped/murdered thousands of women and children?
- In Season 1, the Dothrakis raped/murdered women and children in Mirri's village on a weekly basis. So it feels poetic that in Season 8, Dany would instruct the Dothrakis to kill thousands of women and children on the streets of King's Landing.
- This is only season 1 of the show: Dany is already talking about how much she enjoys hearing screams from people as they burn.
I'm not saying that Dany didn't have a good side or good intentions at one point. And I'm not saying that you should feel bad for rooting for Dany.
But even if Season 8 had never happened, I would still say she was about as much of an "inspiration" as Walter White from Breaking Bad.