It is my name that is more important than any logo.
See my blogs...no logo....never had a logo.
Same with header graphics....
Even might coca cola found out that just having a sign that says COKE is more noticeable than a huge logo sign.
COKE is a trademark, not a logo.
Bottom line is, a logo is not a requirement for a business.
Your logo is the first step to strong branding and every website needs good branding to retain visitors. A strong layout with good content and design can serve as that branding because repeat visitors still know they are in the place they intended to be. But a familiar logo tells them that right away. If a website is redesigned and they don't have a branded logo then repeat visitors can be lost because the branding WAS the old design and layout and the branding dies with it. Carrying over the logo makes for a smoother transition.
I agree that a logo is very important for a business. As soon as you start using social media, you suddenly need various images (facebook pages, app icons, profile images etc.) that the logo is useful for. Many of these images are square in shape, so it's difficult to use only your name, or business name which tends to be long and thin.
Logo is very much important thing for starting any thing. It should be attractive.