According to this
Edwards is now the 31st commonest surname across the UK. A lot of those are in Wales, where names of that type are especially common, but they're also spread across every other county.
To estimate 17th century numbers, divide by 10. So there were about 15,000 Edwardses then. Which means about 500 Edwardses were born every year, and about 50 of those were called John, a dozen Thomas, etc. Ballpark figures.
They weren't all related. Names of that type were adopted by many unrelated families independently. Probably most surnames do have single origins, but (outside Scotland) single-origin names tend to have lower frequencies, and few if any of the top 300 names go back to a single origin.
In fact names like Edwards mostly became hereditary rather late, around 1300-1400 in England, later in Wales. They hadn't had as long to multiply as older names. So we can estimate that the 15,000 Edwardses in the 17th century would be made up of say 300-500 unrelated clans of average size 30-50, and many of those will have become extinct since then.
You'd often get a situation, especially in Wales, where the sons of an Edward in one village, say Edward ap Huw ap Gwilym, would decide to use the surname Edwards, and the sons of an unrelated Edward in the next village, say Edward ap Hywel ap Gruffudd, would also take the surname Edwards, and then you'd have unrelated Edwards families a mile or two apart. Later, they'd forget their ancestry and sign up to the prevailing myth that they were all one big family.
Which all makes the genealogy very difficult. You need property or wills or something to establish relationships.
In the past many Americans trying to trace British ancestors have tended to assume that if you find the right name, you've found the right person, and if you find two men with the same surname 30 years apart, they'll be father and son. And so they cobbled together "lines" that are really just lists of unrelated random people. And those lines are now all over the internet.