Is using a globe valve on a water supply line a bad idea?
A gate valve does not have a "washer set". The valve in the photo appears to be a globe valve. 

I agree with your plumber that a globe valve is proper in that place. A gate valve or ball valve is intended to be seldom closed, usually to empty out pipes for winter exposure or to repair the plumbing. If you must operate a valve often, as part of your household routine, then a ball valve may not stand the wear. 

Some differences among the most commonly used valve types: 

A gate valve stops the flow by sliding a flat plate across the flow path. The advantage of a gate valve is that when fully opened it offers almost no restriction to the liquid flow. The main disadvantage of a gate valve is that the end of the gate must slide into a pocket to complete the seal, and this pocket is not flushed by the normal flow of liquid, so that scale and debris collects there and blocks the valve from fully closing. 

A ball valve controls the flow by rotating a ball with a large hole in it. The advantage of a ball valve is that when fully opened it offers less restriction to the liquid flow than most other types (except perhaps a gate valve). The main disadvantage of a ball valve is that when the ball eventually wears down, there is no adjustment to tighten the mechanism and stop the leaking. The ball type should not be used where the valve must be operated frequently. 

A globe valve operates by screwing a small disk, faced with a rubber washer, directly down onto an opening in a smooth plate, called the seat. The advantage of a globe valve is that it is simple to manufacture and maintain. As the washer and seat wear down, simply turning the handle little further completely stops the flow. Most users do not even notice this. When the washer becomes too worn to seal properly, it can easily (and almost intuitively) be replaced by an amateur. Even a damaged seat can be reground without special training -- just the proper tool. The disadvantage of the globe valve is that even when fully opened it still significantly restricts the liquid flow. 

In your case the flow restriction will not be a problem because you are not using the flow directly in your shower etc. Even if you notice a difference, it will just take a bit longer to fill your tank.