The big cap works kind of like rolling a car that first gear went out off you get a boost to get going. All joke aside it charges up and gives the needed power.
I started doing it with them esp8266. It gives you a bigger supply for a short time. I got the idea from them supper caps.
To be fair, it's normal design practice to include a decent size electrolytic across the supply - not doing so is simply asking for trouble, particularly on anything battery powered.
I maintain machines that have industrial servo's, there are all kinds of issues with these, the problem you mentioned mr doggy happens when the intergral gain is too low, if your using cheaper servo's maybe they are not setup that well.
So your saying this happens when there isnt enough juice to power the servo properly?, that makes sense, the integral gain is usually lower than the proportional, so your effectively removing the integral from the control loop if this is the case.