Everett was added in 1919, almost two decades after Sullivan. It was never intended to be a permanent terminal, just the start of an extension to Malden (or beyond) with an elevated structure over Main Street. It reduced the round trip distance for streetcars running to Sullivan by two miles, allowing some of those streetcars to be reallocated elsewhere - much like why the Roslindale extension is now proposed. Unlike Rozzie would be, Everett primarily served as a streetcar transfer point, as other than the small residential neighborhood on the east side of Broadway there wasn't much around except for heavy industry. Its location was chosen because it was on the grounds of the existing Everett shops, minimizing expensive property acquisitions.
By 1926, the El to Malden was dead because no one wanted new elevated lines. The proposal in the
1926 report was to extend from Everett station over the Saugus Branch to Lynn. Good for Malden and Lynn, but no real improvement for the Broadway corridor in Everett. I don't believe there was ever a serious proposal to tunnel under Broadway, probably because the BERy built everything outside Boston city limits on its own dime.
By
1945, all planned extensions were along existing rail rights-of-way, and the Western Route to Reading was a much better fit than the Saugus Branch - especially since Lynn would be served more directly by what is now the Blue Line. A
circumfrential route from Everett to Boston via Medford was studied at legislative demand, but found not to be needed. (The report also studied
some ways to replace the existing El structure between North Station and Sullivan Square, but recommended the Sullivan Square–Everett section be retained as a feeder.)
In March 1963, the Everett-terminating trackless trolley routes were converted to diesel bus. With ridership down substantially since its peak, and diesel buses able to use the de-electrified stretch of Alford Street to Sullivan, Everett was closed nights and Sundays at that time. It was cheaper to run the buses to Sullivan at that time than to staff Everett station and run rather empty trains there.
When the Haymarket North Extension was designed, the line was trenched through Sullivan Square to avoid grade crossings. (Elevated on the Western Route alignment would have conflicted with the overpass and I-93, and no one wanted to keep the existing elevated.) Without the ability to connect it to the new route, the Sullivan Square–Everett stub would have been a shuttle service, requiring an additional transfer versus simply running buses to Sullivan Square. And so, in 1975, Everett station passed into the dustbin of history.
tl;dr: The industrial location never served Everett well, and keeping it was not compatible with the highly-desired extension to Malden and beyond.