Thanks. I feel I'm getting somewhere. You see nothing you have said has suggested it cannot be done. You have described problems that will be encountered along the way and would need to be solved, not anything that would stop it from happening. I think technology has changed since the old days, with fibreoptics so maybe you can avoid hogging the lines, and of course there is mobile phone technology to consider. For example, I have built a prototype of a similar system that uses sms to turn any mobile into a mini server but as a text is only 140 characters it's of limited use.
So to make the modem system work you would need a modem, a phone line and a modified browser. Oh and the DNS list of phone numbers. That would be the problem as you would either have to know the number, say in the link, or keep a database of the DNS on every PC, or call up a central server that has this every time you need to find the number. That would be an issue. But again any problem can be overcome with clever motivated people.
If it became popular then there is no reason that phone companies would look to supplying the need, such as allowing broadband cables to be used for this. The reason for wanting to do this of course is to try to stop ISPs from being able to monitor what people are doing, or having too much power, so it's political, democratic, but then I guess phone companies would eventually be able to do the snooping. So, perhaps there's nothing to gain from doing it. And cost wise, you would have to pay for each call which may make it pricey. Still it would be fun to make it work.