There are potentially a few issues with either "Convert Premul", converting alpha, or a simple multiply across the channels, that may not be entirely obvious at first. It is a point of confusion in that that no node should deliver unassociated alpha as the entire compositor assumes associated values. Sadly the node used here, does.
One issue, with particular attention to pulling a key, is that your edges will likely not be of the correct luminance value to blend seamlessly. That is, when a subject is shot against a green screen, the alpha channel that is produced would be a ratio of the foreground against the background, which in this case is some non-black value of the screen used. The resulting output ratio isn't going to be the same as an object set against non-emitting pixels. Those semi-transparents can be extremely tricky to get the ratios correct on without having a good grasp on the behind-the-scenes models.
The second issue would be with associating is that you can destroy RGB information. As per the OpenEXR Technical Introduction, alpha with a greater ratio RGB is entirely valid and represents emission with a lower degree of occlusion. The problem is that if you craft this in an RGB image, zero alpha will destroy this information on the multiply.
Calling the color channels “premultiplied” does not mean that the
color values in an image have actually been multiplied by alpha at
some point during the creation of the image, or that pixels with zero
alpha and non-zero color channels are illegal. Non-zero color with
zero alpha is legal; such a pixel represents an object that emits
light even though it is completely transparent, for example, a candle
flame or a lens flare.
In the visual effects industry premultiplied color channels are the
norm, and application software packages typically use internal image
representations that are also premultiplied.
To work around this, a carefully crafted RGB against zero emission RGB values can be associated manually.
Once you have a representation of your emission levels in RGB, which is effectively RGB values against black, you can use whatever technique you desire to assign the alpha channel. For example, if you have a blue glow with low or zero occlusion (alpha), you would want the equivalent occlusion values properly set into your base. "Set Alpha" will assign the alpha directly to the alpha channel, and the result should be correct.
TL;DR in your sample, "Convert Premul", using the alpha conversion node, or strictly a simple multiply will all work as expected, with a rather large caveat being the first issue above. You would ideally want your image to represent emission on the semi-occluded pixels, which as any keyer would tell you, is quite a dance of artistry. Using that canned After-Effects-Esque "Keying" nightmare node won't help either, as it assumes a single luminance / RGB value to compare against across the whole green screen, which is never the case, and will result in whack or rather cheated values, which will impact your final composite. An alternative that might get you closer is to inpaint and use a distance node on the RGB values, which will at least pull data from the fully non-occluded plate elements and push them into the semi-transparent regions. This will result in a luminance difference that is closer to what was behind the actual pixels in question.
 Note that there are largely two primary methods to do this. One is probabilistic, aka A+B-AB and the other is non-overlapping, which is A+B. If you are certain your geometry doesn't overlap for the pixel in question, the latter is correct. If your geometry overlaps, such as a transparent window atop of another transparent window, the former is correct.