You had 3 professionals and there’s still no solution? I don’t know anything about your home / water heater’s location, WH BTU’s, draft hood size, surrounding area size / combustion air supply, chimney type, chimney’s existing liner diameter and type / material, your local jurisdiction code requirements, does the water heater includes FVIR feature…
Those technicians gave you opinions based on observation of the chimney and water heater’s installation; I don’t have that information…
The chimney MUST have a properly sized liner installed / water heater cannot vent into an unlined chimney (in some areas caly liner isn't sufficient either). Your jurisdiction may require certain type of a liner to be installed based on your geographical location (might require insulation) and locally adopted code.
Kitchen, bathroom, and laundry exhaust vents cannot, under any circumstances, discharge into a chimney used for venting of fossil fuel burning appliances.
Chimneys require vent caps and protective screens are highly recommended as well.
Water heater’s vent pipe – you may check the following link for everything you’d need to know about it - water heater vent pipe installation.
However, I don’t believe that any of the above would be responsible for the pilot light to go out because the WH's draft hood should prevent that from happening. Nonetheless, all those things should be corrected, brought up to the current / your local code.
Common conditions that might be responsible for the pilot light to go out:
Thermocouple and the thermostat – it could happen that those new parts (or one of them) have problems but let’s assume that they don’t.
The information about this next condition is here (very probable cause if you have this type of water heater): FVIR safety feature
I don’t know how old is your water heater but effective July 1, 2003, all residential 30, 40, and 50 gallon atmospheric vent (natural draft) water heaters are required to be equipped with FVIR -flammable vapor ignition resistant design.
Next extremely important item is explained here: combustion air requirements
Operating open hearth fireplace creates sort of a vacuum that might cause the water heater’s pilot light to go out if you have a “tight” home. It simply pulls the air through any other “holes” in your home’s envelope. Similar situation could happen while using a whole house fan without opening doors / windows.
However, this probably wouldn’t happen on a sealed burner chamber water heater equipped with FVIR system (if you have one of those).
I can’t think of anything else… do you have a Carbon Monoxide detector installed, any activity? Just click on "comment" below.