I think a runs test is a good idea. To me, by analysing the data in "chunks," your intention is to create a proxy for or control for "hot hands" in player consistency. There's a huge literature on this phenomenon out there. One of the best papers was discussed by Gelman on his blog back in July 2015. The title of his post was, "Hey-guess what? There really is a hot hand!" (http://andrewgelman.com/2015/07/09/hey-guess-what-there-really-is-a-hot-hand/). The paper Gelman reports on is a rebuttal to much of the previous literature insofar as it details the errors made by previous analyses of the hot hands phenomenon. The earlier work focused on overall as opposed to conditional probabilities. This paper posits a new sequential probability model (see the link for a reference to the paper).
One good metric of consistency which should control for differences in, e.g., number of shots taken, is the coefficient of variation. The CV is a dimensionless, scale invariant measure of variability and is calculated by dividing the std deviation by the mean. The problem it attempts to solve is that std deviations are expressed in the scale of the unit under measure, i.e., it is not scale invariant. This means that metrics with high average values will also tend to have higher std deviations than metrics with low average values. So, for instance, due to differences in their average values, measures of the variability in diastolic and systolic blood pressure are not directly comparable. By taking the CV, their variability becomes comparable. The same thing holds for many other metrics such as stock prices, online metrics such as the number of impressions or hits to a web page, and so on.
Thus, the CV can be calculated for many metrics and scale types, excluding categorical information and measures with negative values.