Did the Stoics really want people to feel nothing?

No. The popular image says a "stoic" is a person who represses feelings or endures patiently, and the ancient Stoics' own idea was narrower. They wanted to be free of the passions, a state called apatheia, which is better translated as equanimity than indifference. It is very different from modern "apathy", which suggests inertness.

What is a "passion"? For the Stoic Chrysippus, the passions are evaluative judgements: disturbing and misleading forces in the mind that arise from a failure to reason correctly. The Stoics defined a passion as an impulse that is excessive and disobedient to reason. There were four main kinds: distress (a present perceived bad), fear (a future perceived bad), appetite (a future perceived good) and delight (a present perceived good). Take the Stanford Encyclopedia's example: if I am distressed to learn I have a heart disease, the false thought is that illness is bad; for the Stoics it is dis-preferred, but not bad, and makes no difference to happiness.

So the sage is free of those four, but is not an unfeeling robot. The sage experiences clear-headed good feelings (eupatheia), listed as joy, wish and caution. There are also pre-emotions: reactions that occur without assent, and so can be felt even by the Sage. Epictetus allowed that even the wisest person may tremble or grow pale at some sudden danger, though without false assent.

Apatheia was seen as a byproduct of living virtuously, not as a goal to be pursued directly.