"never heard anywhere Mexico saying "let my people go". "
No one said that. But Mexico had outlawed slavery. And if they took Texas, it would have hurt the slave owners and their dependents.
People love to say how most people in the south didn't own slaves. Yeah? Well most people in Flint didn't work for the auto industry. Funny how the town died when GM left though.
Slavery was the backbone of their economy. Losing it would have been devastating.