Net Promoter Score - https://en.wikipedia.org/wiki/Net_Promoter - is used to measure just this.

> Net Promoter or Net Promoter Score (NPS) is a management tool that can be used to gauge the loyalty of a firm's customer relationships

It asks one question:

> How likely is it that you would recommend our company/product/service to a friend or colleague?

The wikipedia article provides more details including criticism.

NPS doesn't measure future goodwill (or present goodwill for that matter). Rationally it may have some correlation, but it isn't a silver bullet even when understood and used properly (which it often isn't).

Instead you often get people trying to game their local NPS scores; not quite the same problem, but similar.

> they measure sales metrics rather than feelings

The term you're looking for is "brand". Measuring a brand's value is much more difficult than measuring sales, but it's not something new that no company do.

I believe any brand loyalty metric would be helpful if tracked consistently over time. If sales are climbing and loyalty is cratering it's time for the board to step in and ask questions.

The dark side of data rears it's ugly head.

More like the dark side of stupid B-school ideas that should be discarded in the dustbin of history. The data is there if one cares to measure them and incentivize properly.

But why would they favor that? They are intelligently blind.