Stop. Go read wage labor and capital if you want to talk about this. Because I’ve described wage labor from marxist definitions.

Yes, I have read it, and it doesn't follow from anything in there that public sector workers aren't doing wage labour.

Their surplus labor value is not being stolen from them.

How? Do you think in order for exploitation to happen a business has to be private? That's not at all true.

Profit is reinvested for the common good.

If there is profit there is wage labour and exploitation. It may be better than privately owned enterprises for a number of reasons but it is still very much wage labour.