There are a lot of arguments around why you should use webfonts. In none of those arguments, have I heard about a single problem being solved for users. And well, I'm only really interested in solving problems for users.
User-testing is an important part of the design process. It allows you to both hear what users have to say as well as observe how they use a product. It can help you separate your ego and desires from the problem solving at hand. Over the last three years I have been fortunate to have participated in a decent number of testing sessions.
In that time I never heard a user complain about:
On the flip side I:
These points were enough to convince me that webfonts cause more problems then they solve and weren't worth the cost to my users or myself. Webfonts come with a number of costs. They cost money to license. They cost time to implement. They cost time to download, and with some mobile data plans they can cost money to download as well.
Typography is not about aesthetics, it's about serving the text. If even a small percentage of people don't consume your content due to a use of webfonts, your typography is failing.
All this being said I care deeply about aesthetics, and I've found the following two sentiments to be true:
System fonts can be beautiful. Webfonts are not a requirement for great typography.