Will manually removing nicks from compressor blades significantly affect the operation of the compressor?
Yes, this is not only acceptable, it is required by the engine maintenance manual. The tools might be more sophisticated in other shops, but the basic concept of removing material by hand is the same.  The process is called "blending". The impact of sand or dirt on the blades causes little nicks and scratches. Those sharp or rough areas create what is known as a "stress concentration".  If left as-is, the nicks will eventually grow into cracks (metal fatigue). Smoothing out the nicks removes the stress concentration and prevents the crack from forming.  There is an impact on the performance/efficiency, but that's better than having your blades crack.  From my experience, the performance impact of what is shown in the video is actually quite small. As a rough guess I would say 0.01% - 0.1% efficiency loss. Much larger nicks can be successfully blended.  

Edit:
Regarding how I came up with that efficiency number. Two thoughts:

First, The compressor shown in the video is a 10 stage compressor, and as a rough estimate, each stage probably has about 30 - 50 blades. I didn't try to count exactly but let's just call it 400 blades. It is a reasonable assumption that each blade does the same amount of work.  In reality some stages will be a little more heavily loaded than others, but it's fairly close. So if you removed 1 entire blade, as a rough order of magnitude estimate, you would expect that to be about 1/400 = 0.25% efficiency loss.  The amount of material removed is clearly much less than an entire blade, so the efficiency loss should be less than 0.25%. 

Second, I have seen compressors that had much much larger nicks be blended out and have on the order of 0.1% - 1% efficiency loss.  

So between the two of those ideas, I made an educated guess on the efficiency number. Could I be off by a little bit?  Absolutely. But I'm probably at least close.