This is an honest question, not an easy question, but an honest one. Finding an answer will take more than just asking a bunch of people and polling the results. Gathering results like that would get you data, but would it answer the question? There is a big difference between an answer and just having data. You also have to pay close attention to many variables and still you may only end up with a small part of the eventual answer. There are a lot of reasons, but the bottom line, science isn't easy to do. It's also not always easy to explain. Even trying to answer a question like this can show you that it's really tough.
Supposed you went to the Discovery Institute and polled people who worked there, what do you think the answer would be? Their poster, Kirk Durston, would certainly tell you everything that he thinks is wrong with science, real and imagined, and certainly describe any negative until it's insurmountable. Of course he doesn't offer any suggestions, his focus is to make people scared of science.
I came across an interesting article, "Science Isn’t Broken" by Christie Aschwanden. It unflinching looks are some of the real problems within science and, more than Kirk ever did, explains how and why some of them occur. She also goes into some of the changes that have been happening to help improve science and scientific methodology. It's a long article and one well worth a read. It repeats something that anyone who is objective about science realizes, that while it's not perfect, it s a self-correcting activity.
On the opposite side is an article on Evolution News and Views which grabs one of the examples Christie Aschwanden discussed and tries to turn it into some backhanded support for Intelligent Design, "Some Scientists Say Intelligent Design Isn't Science -- Until They Have to Use It Themselves." One line really caught me:
That's a lie! What almost all scientists say is that Intelligent Design is not science, they do not say that intelligent design isn't science. Intelligent Design (ID) is a movement, a modern re-telling of the Watchmaker Analogy. ID is not the same thing as intelligent design. In all honesty, if we built buildings using Intelligent Design, I doubt they would hold up for very long, even if they could stay up long enough to be finished. Think about it, let's use concrete that performs based on the wishes of a deity instead of the intelligently designed by real science formula and processes that make the many types of concrete we use in construction."Many scientists claim intelligent design is not science"
Human beings design things all the time, some intelligently and some less-than-intelligently. Many things work, but then I drive past a car jacked up 36 inches in the air running on 14 inch tires at 70 miles per hour on the freeway and all I can do is ask "What were they thinking?" Architecture and Engineering run on intelligent design, but it has very little to do with Intelligent Design.
It's really funny how whoever wrote this one tried to twist around the idea of intelligence and design into an effort to support Intelligent Design. There was no identified author, which isn't too uncommon. I do wonder why no one takes credit for this one. But aside from that, do they really not understand the difference between intelligent design and Intelligent Design?
Seriously? The DI doesn't recognize the difference between something intelligently designed and their teleological argument for the existence of God? Science works, the only thing intelligently designed about Intelligent Design is the marketing campaign. Honestly, it should be taught in marketing classes, not so much as a success at selling their ideas, but from the point of view of how well-funded nonsense can do some actual damage to science and science education.
So back to the original question, is Science broken? No! It's not perfect, it has issues, but what doesn't, religion? it's going to take people interested in improving science to fix issues as they happen. What they will certainly do is intelligently design processes and changes to help deal with issues within science. But just because these changes will be intelligently designed doesn't mean that they will use Intelligent Design. What doesn't help are organizations that inflate real issues to the point they sound catastrophic so they can try and wedge their religious ideas to take the place of real science.