Good Afternoon! Hoping someone can help!

My car is giving me the "Engine Overheating" warning after getting an Oil Change and a Coolant Flush/Exchange.

Here's the Details...

Went to get an Oil Change, while there the Rep recommended a Coolant Flush / Exchange (Said it only had water in it and needed the service)

I agree to it and off I went. Maybe less than 30 miles later, my car gives me the "Engine Overheating" warning, so I pull over to a gas station.

I check the reservoir after it cooled down and it seems low and near empty. I added a bottle of Coolant, cranker the engine and all seem fine. Thermostat needle working and everything.

I drive maybe 3 mins and the Engine Overheating light comes on again. So I pull over to another gas station. Same thing. This time someone said burp the system (Because maybe they forgot to). So I squeeze the hoses, open the reservoir cap, etc. Add coolant and off I went.

But now this time, I see the thermostat not moving and it's stuck on H / red territory after I drive it again.

I get it home and look under the hook and it seems like some coolant is leaking in the bottom but I dont understand how.

Did the mechanic not close something when he did the Coolant flush? Because the car and thermostat was working fine before he did the flush. Now the car is overheating and the thermostat needle just sits at H/ Red territory.