Web developer especially need to get their sea legs and be comfortable working on unstable ground all the time - it's inherent in the kludge we call web standards. So if you take a list of all the things a web developer needs to know to be effective, let's just agree "high math" isn't going to be near the top of that list.
How often would I use this level of math? I know I use the others on an almost daily basis. Coming up with something I need linear algebra for, though? Few and very far between. Maybe you use linear algebra often, but most of us don't have a need for it.
And if you disagree, then I think you should actually know abstract algebra, number theory, topology, etc., so that when the next problem comes along, you can 'quickly come up with an optimal solution'.
"I have been walking through mazes my whole life with my eyes closed, and I make progress by feeling the walls with my hands. In order to get through each maze I need to remember lots of facts about how the walls in the different mazes feel when I touch them. If I opened my eyes now, it would take too much time, and be too painful to adjust to the light. I never tried to open my eyes, and even though a lot of people keep telling me it's a lot easier to get through mazes with your eyes open, I think they are wrong. Besides, the few and far between times I get stuck, I can always ask one of the people who can see where I should go next. And if you think I need to keep my eyes open, then the next thing you are going to tell me is I should use my sense of hearing and smell too."
If you must reword and put his entire argument into a bad analogy, try this one:
"I've been walking through mazes all my life. This one time, I had to recognize a particular brand of poison flower, but I didn't know my flowers very well because flowers are rare in the maze."
And the answer?
"I asked my friend who was also in the maze: 'Is this flower poisonous?' He said yes. I said okay. We skipped the flower and moved on."
If the majority of someone's engineering involves concepts and tasks that one is 100% knowledgeable about, I'd argue that that person has reached a stale state and shoudl probably seek out other challenges.
Specifically, I'm NOT saying that math is not important.
I'm saying that there's nothing at all wrong with getting by with imperfect knowledge in a domain if the process of "getting by" means expanding your knowledge in that domain.
Everyone learns on the job. I don't see the difference between brushing up on linear algebra from school, and learning that new framework or language or software suite.
Second of all, like you said, you can get the "hack" done without math. I would say the less hacks you have to use in your programs the better, and the best practice is to code at a high enough level of generality that you don't need any hacks at all. Without math, so much time is spent (or wasted) tinkering around, and you don't often end up with the optimal solution. Whereas a strong math background lets you skip all that tinkering, because you are able to properly plan a solution before you even write a single line of code.