I am reading into single stage microwave amplifiers and how to design for a specific gain (source 1, source 2 warning PDFs!). In both sources the claim is made that the highest bandwidth is achieved with the smallest reflection magnitude. But why? What is the relation between the magnitude of the reflection coefficient and the bandwidth? I believe I have seen this claim before for matching networks in general, and not just limited to the context of microwave amplifier.
As your signal propagates through the transmission line, when it hits a discontinuity, it will reflect and start heading back. If you have multiple branches in your transmission line, each time you come to one of those intersection, you'll have part of the signal travelling in each of those branches and then back (if the branches are not terminated properly).
A signal travels at a finite speed, and if you have a fast frequency signal, it may change faster than the signal can propagate through the line. So now you have your signal interfering with its past self (reflections) which will then distort your signal.
By minimizing your reflections, you minimize the amount of interaction between the reflected signal and the incident. This then decreases your ISI (inter symbol interference), which allows for more data to pass through without issues and therefore a higher bandwidth.
Here's an intuitive way to look at it. Smaller reflection coefficient can mean less reactance, or lower Q. Less reactance means less frequency dependence and lower Q generally leads to higher bandwidth.
Keep in mind, this is not a hard rule. It is possible you might end up with more bandwidth by matching to a gain circle point further from the center of the smith chart. It really depends on your specific matching topology. Your lecture mentions this by saying, you need to test it to know for sure.