Your thinking process is exactly right. And you won't find the explicit result for the test "No Effect vs. Location and Scale", which states the null hypothesis that scale and shape parameters are the same.
That test, by itself, is insufficient to give additional insight if the null is rejected. And additional tests are needed. In that case, it will end up with a nested test. Let me use this example to explain why nested test is useful for decision making.
In this example with two groups, one is usually interested in whether their reliabilities are same, or one is more reliable than the other, so one can take proper actions accordingly. When both tests come back insignificant (denote this Scenario 1), then one can conclude that their reliabilities are not different, then take an action. If "No Effect vs Location" is significant, but "Location vs Location and Scale" is insignificant (denote this Scenario 2), one can conclude that one is more reliable than another other in the conventional sense, then take an action. In the remaining two scenarios (denote them Scenario 3 and 4), one can only conclude that the two groups are different, but cannot draw conclusion on whether one group is definitely more reliable. A further analysis is required, or the criteria of being more reliable needs additional specifications, e.g. defined by comparing B10 life.
So what about the "No Effect vs Location and Scale", which is not there. But think about it, if it cannot be rejected, it means Scenario 1; if it is rejected, it means one of the other three scenarios, but cannot tell which. So the nested test can directly lead to decision making about the most common interests, but "No Effect vs Location and Scale" cannot.
P.S. though I describe the decision making process as a multiple testing, it is often like a sequential test in practice. I.e. it seems rare if "No Effect vs. Location" is insignificant but "Location vs Location and Scale" is significant. So one may describe two groups are no difference just based on the first test without further elaboration. Also if you do want the test result for "No Effect vs Location and Scale", it Likelihood Ratio Test statistic is 311.6156 - 310.3634 = 1.25219999999996, the difference of -2Loglikelihood from the two models. And the critical value is 1-alpha quantile of Chi-Sqaure with df = 4-2, the difference of the Number of Parameters from the two models. The p-value is 0.534672964662089, using the following calculation:
Ref: https://en.wikipedia.org/wiki/Likelihood-ratio_test