Top rope belay with ascender?

I saw some folks top rope belaying with an ascender, in addition to their normal Grigri, purely for convenience of pulling down rope, so far as I can tell.  Does this provide other benefits?  Is it really that helpful?  Is there a downside to this approach?  

It is a method used if you need to pull on the rope harder. Had it done on me for a mixed route so I would not fall as far due to rope stretch/extra slack. Could also be used if there is a ton of rope drag (if there is that much drag you probably have an issue thought). It definitely is not needed 99.99% of the time. 

Solid thumbs up. For long belays, projecting, or drag as Will mentioned it's really convenient to have a jumar. No you don't absolutely need it but it's usually in my pack when I'm cragging. 

The main downside is that it becomes unwieldy if the climber is moving really fast, so I don't use it on easy routes. Also you'll inevitably forget it on the rope when you lower the climber.

El Duderinowrote:

belaying with an ascender, in addition to their normal Grigri, purely for convenience of pulling down rope, 

As you noted in second part, not belaying w ascender. Yes, it’s very convenient for long single pitches.

Does this provide other benefits?  

Do you also wish to climb the route? If so, your shoulder(s) will probably thank you. Big Red, going to disagree about faster climbing, as it allows for really responsive belays due to being able to easily pull in 2-3ft of slack with little effort. Important when your partner is barely taking 4-5min to do a 130ft pitch.

  Is there a downside to this approach?  

Aside from being mindful of the additional equipment you introduced to the belay process, not really. Watched Jacopo belay Babzi this way on Ahab last year, wasn’t the first time I’ve seen it. 

Long TR pitches like 30m+ with tons of rope stretch is where it comes in handy, especially if ice climbing and belaying with big heavy gloves. Saves a surprising amount of energy from not having to pull on an icy rope for half the day. Any sort of TR where you want to keep the rope pretty tight (beginners on crack or ice, etc.) it's also helpful. I don't bring it rock cragging typically but almost always have one ascender in my group going ice cragging. 

I have done this occasionally. It is most helpful on the longest top ropes. There are some 50-60meter routes here I like to top rope by using two ropes tied together. The handle grip on the ascender is much more ergonomic than gripping the rope. It is nicer on the skin (if not wearing gloves) and the wrist. It’s less work on the hands, arms, and shoulders as mentioned above.

Not all climbers are strong. Some have arthritis, joint pain, tendon pain, weaknesses attributed to advancing age or any variety of valid reasons to appreciate tools that help make belaying easier. 

Proper P.B.U.S. belay technique is more awkward with the ascender involved. Modifying with the less proper “tunnel hand slide” is an alternative; acknowledging that it is less secure. However, I find in practice I can hold the weight of a climber in my hand with the ascender while my other hand catches up to tension. Usually the longer pitches create higher friction, lessening the required holding power of the belayer.

Very useful in belaying climbers extremely tight on the rope if for some reason they would want that.

Useful for overcoming that pesky high friction top rope all belayers dread. Of course, learn to avoid/mitigate setups that cause so much drag in the first place, but the ascender does help belay easier, either way.