The HVAC industry has a pretty extreme mark up on parts and the service they provide. You are pretty much at the mercy of the service provider unless you know how to work on these machines, The best you can do is shop around and see if you can get a slightly better deal. I taught myself the basics when I bought my first home and have never looked back. I would change it myself, but not everyone is not capable of doing so.

I think new is better than guess and replacing.

the tech took this picture from the attic he said the evaporator coil is leaking oil And this whole evaporator coil has to be replaced. Can someone validate if this is a known problem or fak?

Welcome to home ownership. Air conditioners don't last forever.

If you don't believe it this picture (below) maybe thought to be a fake as well.

Belief is a funny thing this day and age.

I'm sorta like Mike (wasn't there a campain button along that line) ,unless properly zoned,5 tons only make's sense when multiple people are home 24-7. Even in my case of 2 people home 24-7,I have 2 units in both homes.

Oil on coils is an indicator but not a certainty of a leak. I would expect a leak test be done.

I’m planing to live there probably for more than 5 years. The house was built in 4/2010. Thus, this AC unit is 10 years old. I contacted the original company that installed the unit and they said the AC 10 years warranty just expired 4/19/2020. So the total repair the tech quoted me $1750 (this included replacing brand new coil and labor). Is this price reasonable? He did perform a leak test by infusing with nitrogen gas to hear the leaks.

You've gotten probably halve the life out of this system. It would be a hard choice to just replace the coil when you may have to replace everything (maybe) in 5-10 years.