Oh boy there's a number of inaccuracies here but I'll start off with the main question of why the Bell System wasn't broken up until 1984. In short, it wasn't for a lack of trying. Post WWII, it was already recognized by the Justice Department that the Bell System was engaged in anti-competitive behavior. This stemmed from both the vast physical coverage of AT&T and its component companies, and also the fact that AT&T fully owned Western Electric which was its manufacturing arm. They were engaged in a number of practices such as requiring exclusivity deals with independent telephone companies (of which there were and still are thousands) when it came to purchasing equipment.

In 1956 they entered into a consent agreement with the DOJ which enjoined them from doing a lot of things such as selling equipment outside of the Bell System and from making equipment that wasn't intended for use in common carrier systems. Now that's a very broad definition, but one of the big things it did was to stop them from getting into the nascent computer industry. However, the 1956 consent decree could be broadly interpreted, and it still didn't have any effect on the broad reach of AT&T.

By the 1970s it was clear that AT&T still had a virtual monopoly over the US phone system including building the vast majority of equipment that was used. There were, of course, other companies that made equipment (Automatic Electric, Stromberg-Carlson, and a smattering of smaller companies most people have never heard of) but they were often limited to selling to the "mom and pop" independents which were more often than not rural and small. This is what eventually lead to the filing of an anti-trust suit in 1974 in an attempt to break up these companies.

This case took years but AT&T came to the conclusion that it would not end in its favor. In 1982 they agreed to another consent decree with two key components: the regional Bell operating companies (RBOCs) would be divested into seven regional companies, and there would be no requirement that they purchase from Western Electric. In return the 1956 agreement was dropped which gave AT&T the ability to enter more markets.

So in 1984 the seven RBOCs split off into their own companies. This was a boon for other equipment manufacturers such as Nortel and Siemens who now had access to a market that was previously mostly unavailable to them. It also allowed AT&T to enter markets that they hadn't previously been in such as computers. They purchased NCR in 1991 for instance to attempt this, although their success here is still up for some debate.

Now to address a couple misconceptions. It's debatable whether AT&T's monopoly status had much of an effect on the uptake of the Internet outside of academic, government, and some business applications. The Internet as we know it started in 1969 and by the 1980s was pretty widespread, just not in use by residential consumers. A lot of this was due to technological limitations. Consumer grade modems were extremely expensive and very slow. Technology like DSL was still far off. While Bell Labs had been working on higher speed technology over voice lines for quite some time, the equipment was generally bulky and not something that residential consumers would be able to install, use, or afford. There was no concept of Internet service providers during this time. The Internet back then was seen as mostly useful for research institutions to communicate and for military purposes.

That's not to say that modems weren't widespread. The Bell System had been developing modems since the early 1950s. The first were built to tie together the monitoring stations for the SAGE system which was developed to detect incoming missile attacks. There were hundreds of models of modems that AT&T produced. They just weren't really marketed to consumers. I suppose it was a chicken and egg problem. There wasn't a push for consumer grade modems because there wasn't much to do with them for the typical user. There wasn't much to do with them since there was no one to drive demand for services.

By the late 1970s computers started entering the home market due to the proliferation of integrated circuit technology. Recall that Intel's first processor, the 4004, didn't come out until 1971 and it was mostly useful for powering calculators. The technology still needed time and it wasn't until 1975 when the 6502 was introduced. This ushered in things like home gaming systems and various Apple and Commodore computers. Intel released the 8086 in 1978 and the PC revolution really started.

Now as far as connecting foreign (non-Bell provided) equipment to telephone lines, that was also a long and drawn out process. The Hush-a-Phone ruling in 1956 held that mechanical connections to the phone network were allowable. If you've ever seen old movies where people take a phone handset and put it in a cradle to communicate over the network (an acoustic coupler), that's a mechanical connection. The next big step here was the Carterphone decision in 1968 allowed electrical connections to the network, with some guidelines to protect network equipment. AT&T developed a number of interconnection standards to support this. So connecting equipment to phone lines was allowable well before the 1984 divestiture.

The aforementioned developments in microchips both allowed for more affordable and compact modems. More accessible home computers gave people the ability to start bulletin board systems (BBSs) which catered more to the home computer market. Things like commercial ISPs weren't really blocked by AT&T for the most part as that really didn't take off until the early 1990s and even then it was nothing like today.

So I think there's not a specific answer for why it took so long. But instead one has to look at the progression of attempts by the government to limit and break up the Bell monopoly over many decades.

Sources:

Disconnecting Parties: Managing the Bell System Break-up -- An Inside View, W. Brooke Tunstall, 1985

Vertical Integration and the Communications Industry: Separation of Western Electric and AT&T, Manley R. Irwin, 3 Antitrust L. & Econ. Rev. 125 (1969)

United States v. Western Electric Co. and AT&T Co., Civil Action No. 17-49, NJ Cir., 1956 Trade Cas. (CCH) P68,246, Jan. 24, 1956

United States v. AT&T et al, Civil Action Nos. 74-1698 and 82-0192, DC Cir., 552 F. Supp. 131, August 11, 1982

Hush-A-Phone v. United States, DC Cir., 238 F.2d 266, 1956

Carterphone Report and Order, FCC, 13 F.C.C.2d 420, 1968