having a low level understanding of program flow can help in many situations. If you are an extremely lucky programmer, you might be able to live your life only in high level languages, but most that I have met had to get down in the plumbing eventually. Understanding asm will help with that.

I am not so sure. To be a good web developer, you need to have a familiarity beyond css, js, html, and the framework you are using. You should be able to understand the underlying protocol, for example, what do the HTTP headers look like? How are they separated? How can SQL injection be dependent on header contents? You can unknowingly build a vulnerable web app if you don't understand the mechanics.

And in my mind, "serious about programming" extends beyond building web apps.

I disagree. I recently designed and implemented virtualized image servers (real ones coming once the Requisition request comes through) for faster image serving. This was basically unheard of by anyone I work with however by having read the http specs I was able to give a report on why we wanted to serve images in parallel. It's easy to dismiss Web Developers as (X)HTML/CSS, Javascript & Php. In reality there are FAR, FAR, FAR more uses for programming competent Web Developers. Shoot, even having a good grasp of fundamental Computer Science helps with understanding Php's array's (or lack-thereof ;) ).

Now for practical usage? Sure Assembly is probably a waste of time. I still haven't found a real way to implement that wouldn't be. But the fundamental understanding of how a computer works, how decisions are made so close to the machine enhanced my understanding about programming which in turn enhanced my understanding about how I wanted to develop and engineer, even for the web.

My .02 cents anyway.