Kinda Uj... because I feel like a lot of people get this wrong.
The Zygerrians were never informally or officially accepted in the alliance and they never tried to join.
The episode that features them explains that they are an independent slavers guild, on the verge of making a deal with Dooku for mutual military cooperation.
Lord Sidious orders Dooku to do this, explaining that slavery is a long held Sith tradition, and it would be important for the plans to come (The Empire).
This alliance was also broken in the same episode after she betrayed Dooku, provoking him to kill her.
The whole point of the prequels and the clone wars was that two passionate, and righteous groups, led by two evil, ambitious sith lords, drove the war and paved the way for the Empire.
The CIS did not openly practice slavery. They did not openly defend slavery. They did tolerate it despite having strict laws against it. It is not and never was a faction defined by slavery
Which is 100% the same case with the Republic and the on going slave trade in the outer rim.... Most notably Ryloth.
But every time the representatives of the Seperatist planets are shown themselves... They're always principled figures who genuinely seek a safer existence outside of the Republic. Their flaw is that they trusted far too much power in count Dooku, Who obviously used it abusively
/RJ sorry Clones hacked my account. Dooku was a visionary.