I say yes and no. There are millions of pages out there working fine without a special design.
Pages people made for themselves, friends and their family. As long as THEY like the look they can call it their own design. What you and I think about their pages does not matter.
That's because each person on this earth has their own opinion about how things should be and look like. You can not find two of the kind and therefore it will be just as many designs as it is living persons creating pages on the web. So it is, and so it will be.
You can of course change a person's first natural feeling in many different ways, good or bad. A good one is education. It could be about anything, and I'm sure you are an expert in some area that gives you more "feelings" then others who knows less in that subject.
Take wine. A beginner knows nothing, all wine taste almost the same. After a while testing and some study in the subject, they will find big differences in the taste.
The bad ones, is bad influence or pressure from other peoples opinions.
Do not ever buy a design that you do not like to 100% because the agency say it best for your business. They only say so because they can not deliver what you really like as they do not have the right skills or education to do so.
As we all have our brains in a left and right part, some people (strong left) do not have the ability to find ideas for their web design and need help from an agency.
(The concept of right brain and left brain thinking is developed from an American psycho-biologist Roger W. Sperry. simultaneous way, looking first at the whole picture then the details. The left brain is verbal and processes information in an analytical and sequential way, looking first at the pieces then putting them together to get the whole.)
An artist or designer (strong right) can actually "see" the painting or the design on the white canvas long before they start painting. On the other hand, they have so many options to choose from, so they need a lot of input from their client to find out what he would like to see when all the parts come together as a website.