Unexplained overheating problem

The problem started 3 weeks ago.

On the passenger side there was some burnt smell while driving and it was intermittent.

The heat gauge was at the center and indicated no overheating. I constantly watched the parking space and found no stains or puddles.

I opened the hood and kept the engine running, there was no signs of coolant spill or anything like that. But then the could get the burnt smell whenever the radiator fan switched ON. The smell died whenever it switched off automatically.

I wasnt using the heater since the alternator was bad.
After changing the alternator, switched on the heater and cold air was blowing. I came back home and parked the car and opened the hood to see a trace of coolant on the air intake duct (right next to the coolant reservoir or the hood-opening latch.

The next day on my way to office the heat gauge went to full and I parked the car and opened the hood to see that it had just started smoking on the inside and coolant had spilled out from the radiator cap side(The cap was intact) and it was white.

I got it towed to a mechanic and they did a pressure test for any leaks and found there were none. They filled coolant and the kept the engine running for 20 minutes and the gauge stayed right on center. It didnt pick up instantly either and behaved normally ruling out thermostat issue.

The fan switches on and off correctly.

The mechanic suspects it could be a water pump problem and have given estimates for replacing it. They are not sure though.

The car had a water pump and timing belt replaced a year ago and has only 3500 miles on it.

After this incident I have driven the car for 5-6 miles at a stretch for a couple of days without any overheating after they filled the coolant. I also didnt notice any loss in coolant in this limited time.

Could someone please advise me on what I could do next.