Since you're a resident-expert on the matter, mind if I ask how much of the SB-era Rams started with Vermeil and where did Martz take it? As a snap reaction I'd always though Dick was too quickly forgotten when it came to the Rams' success. Am I mistaken?
Personally, I think it very much started with Vermeil, though that's somewhat mixed. The foundation for the Greatest Show teams was the player acquisition from 1997-1999. With Vermeil having the final say in the FO, they drafted Orlando Pace, Dexter McCleon and Ryan Tucker in '97, Grant Wistrom, Robert Holcombe, Leonard Little, Az Hakim and Roland Williams in '98, and Torry Holt and Dre Bly in '99. They also added Warner, of course, and signed London Fletcher. All those guys were starters in Super Bowls, and Vermeil deserves a lot of credit for those decisions, and having the right guys in place to help him make those decisions.
On the field was a bit different. Vermeil made some mistakes with the team early, and had to make a lot of adjustments. He started to lose the team towards the end of '98, and things had to be shaken up. Thus, Mike Martz becoming the OC, and some more good player acquisitions in Trent Green and Marshall Faulk. '99 was a perfect storm resulting from the adjustments made after '98 being good ones, and a lot of the moves made before that hadn't paid off yet coming due as well. As soon as some of those things became apparent, the players bought back in very quickly.
Also, Vermeil was an excellent gameday coach. He managed the clock, kept things under control on the sidelines, and the Rams always seemed to make halftime adjustments that worked. He kept things calm and under control, and the players were disciplined.
When he retired, two key members of the coaching staff immediately quit because they couldn't get along with Martz, including the co-DC. It became obvious pretty quickly that another thing Vermeil had been doing was keeping the tendencies of his coordinators under control, as Martz started going insane with his offensive risk-taking, and the former co-DC now full DC started adding so many layers of complexity to the defense that nobody had any idea what was going on.
Martz was more brilliant than Vermeil, but far more erratic. He's one of the worst gameday head coaches ever. He often refused to make adjustments that were obvious, he blew timeouts and made weird decisions with challenges, and often took so long to get plays in that he eventually had to give up the primary playcalling duties, which was originally his strength.
He was an enormous part of the success under Vermeil, but he just wasn't suited to being a head coach, and managing a staff of different personalities, or evaluating players.