Absolutely! Software is a huge multiplication factor. No argument there!
However the point, which maybe I didn’t make clearly, is this: they expect a specific multiplication factor from a specific bit of code. This is the value X they hope to achieve by employing you. If you can do X in less time/lower overall cost then someone else, then that is worth paying more for. However, that X benefit they are seeking is relatively fixed. As a consequence your value is fixed.
With management, there is much more variation.
A bad manager will make a team of 5 produce the value of a team of 3. A great manager can take that same team and get a value of 8. This scales with how many people you manage/lead. Assume management makes a 10% difference to output capability. If you are leading/managing 100 people, the delta value is 10 whole people paychecks.
To bring this home: think how productive you were under a “bad” manager vs a “good” manager. Whatever delta was there, you can multiple it by your normal team size. That is the “value” a good manager unlocks.
I still don’t get why you think that the budget for sharing productivity gains is strictly more bounded for software engineers than managers.
I could flip your example around. McDonald’s hopes to achieve value X from each of their managers and that’s what their willing to pay. But a software engineer who can increase retention rates can noticeably increase the value of the whole company. And that’s part of why one gets paid more than the other.
Modern software engineer is in most cases just a bricklayer - he just realises a vision of the business (product owner and his boss etc.). So, in your example, it's not the software engineer who increased retention rates, but rather the manager who thought of writing such improvement, and hired an engineer who then implemented it.
In other words, the implementation is seen as a trivial part of the process, while figuring out what exactly to implement is seen as the core of the value creation.
However the point, which maybe I didn’t make clearly, is this: they expect a specific multiplication factor from a specific bit of code. This is the value X they hope to achieve by employing you. If you can do X in less time/lower overall cost then someone else, then that is worth paying more for. However, that X benefit they are seeking is relatively fixed. As a consequence your value is fixed.
With management, there is much more variation.
A bad manager will make a team of 5 produce the value of a team of 3. A great manager can take that same team and get a value of 8. This scales with how many people you manage/lead. Assume management makes a 10% difference to output capability. If you are leading/managing 100 people, the delta value is 10 whole people paychecks.
To bring this home: think how productive you were under a “bad” manager vs a “good” manager. Whatever delta was there, you can multiple it by your normal team size. That is the “value” a good manager unlocks.