TL;DR/Abstract - Political, financial, and social expedience are often the dominant forces in the development of infrastructure projects - Sea-Tac is no exception.
The history of Sea-Tac is messy, as airports in urban areas always are. The short version of the answer is that it's a combination of sociotechnological-lock in, prohibitive construction costs, NIMBY/BANANAism.
For more in-depth blow by blow of the third runway expansion I'll point you to this article on the subject, excerpted below.
Most people are familiar with NIMBY (Not In My BackYard). That force has only gotten stronger since the 60s/70s when the term became popularized. The planning field has begun to use BANANA instead: "Build Absolutely Nothing Anywhere Near Anything."
Stuff like airports get built as far from population centers as is economically feasible, and as close to them as is politically feasible. In Sea-Tac's case, they had political tailwinds: World War II
On March 7, 1942, three months after Japan attacked Pearl Harbor, the Port of Seattle undertook development of Seattle-Tacoma International Airport at the urging of the federal government. A new airport was desperately needed to help relieve pressure on Seattle’s Boeing Field and Tacoma’s McChord Field, which were commandeered for warplane production and military use in World War II. After rejecting a site on the Sammamish Plateau as unsafe, the Port chose a sparsely populated plateau near Bow Lake, roughly midway between Seattle and Tacoma.
The 2nd runway went in as part of the '67-'73 airport expansion, and at that point you've got a question of where to put the thing. Note that they decided to put it between the terminal and the existing runway. We can look to the history of the third runway for our answer of why:
The third runway (Now 34L/16R), which was explicitly constructed to compensate for the weaknesses of 16L/34R, required (emphasis added):
...the westward extension of the airport plateau atop some 17 million cubic yards of fill dirt, ... The Draft EIS identified the need to acquire about 400 homes and cited potential effects on nearby creeks and wetlands.
The third runway required legal action all the way up to the Supreme Court of the State of Washington - a $15M, 10 year legal effort on the part of the project.
By comparison, runway 34R/16L was built essentially within the existing boundaries of the airport, which requires almost no permissions or permitting whatsoever.
Mild disclaimer: my doctoral dissertation work focused mostly on energy policy, so I'm extrapolating a little bit here to the transport sector, but given the history of runway 34C, I'd bet money I've got the right of it.
One of the more pernicious forces against change is the presumption of expertise given to experience. "This person's been doing this for forty years," is often viewed as a reason to trust the judgement of a person. Not entirely without reason, mind you, but when technology is advancing rapidly (as was aviation in the 60s/70s), you can run into difficulty where the person who's been building airports for forty years (since the dawn of aviation at that point) "knows" how they should be built and what minimum distances you need.
The issue is that most of that experience is in building airfields for piston-driven stuff like the DC-3 and DC-6. Jet aircraft were rapidly taking over the fleet, and whoever Sea-Tac was allowing to make decisions about runways was demonstrably behind the technological curve: the main runway (34C) had to be repeatedly lengthened as each successive assumption about how long the runway would need to be proved incorrect - and this is in Boeing's backyard. We see this effect time and again in sector after sector from city planning, to medicine, to electrical power: "This is how it's always been done" is an incredibly powerful rhetorical tool.
So at the time, 800ft. lateral separation was probably billed as "plenty." After all, the newfangled jets were expensive, how much of the air transportation sector could they possibly come to claim? 34R/16L was longer than the 1st runway by far, and so presumably was there for the 'rare occasion' when you'd need to land something big.
Ultimately, Sea-Tac had to build the third runway, which involved titanic landscaping, protracted legal battles, and 1.2 billion, with a B, dollars. It also involved several eminent domain actions, and a mountain of environmental impact abatement. Anyone still on the fence in the face of the two forces above can be nudged off by the comparative price tags of the projects.