Q:

Why is there a comma before but in the following case (dependent clause)?

The following example ends with a dependent clause, so I assumed I shouldn't put a comma before but:

She was beautiful enough to attract men but not to intimidate them.

I checked similar sentences on this site.
It seems like there should be a comma in situations like these. Why is this?
Explanation by Grammarly.
Update:
Another example:

He wore a long-sleeve shirt that was too big for him(,) but in a fashionable way.

A:

Great question.
The examples in the Grammarly explanation do make sense.
However, in the other examples given in your list, the included comma reads a lot better than without.

I like him, but not to win.  (timesunion.com )

Imagine reading the above without the comma. It sounds odd. If read too quickly it might even sound like "I like him not to win," which has a totally different meaning. The comma serves to insert a momentary pause in the sentence which reads a lot more naturally.
Another example:

Maybe the idea of a steaming pool of therapeutic water carved out of volcanic rock and heated with underground energy was old hat to him, but not to us. (abcnews.go.com)

Again imagine reading all that without the comma. It's absolutely confusing because there are so many ideas in that one sentence. The "," adds a necessary pause, giving weight to "old hat to him" just prior, to be contrasted with the closing "but not to us". 
All that nuance would be lost without the humble comma!

