Williams appeared over the line for the role on Tuesday night after Richardson released a statement through Port Adelaide saying he had no interest in the Saints coaching position.
However, Williams will now remain Richmond's head of development after the Saints' coaching sub-committee made their recommendation to the club board at 8.30am on Thursday.
"There's been some conjecture that we might have changed our mind during the process – not at all," head of football Chris Pelchen said on Thursday.
"We were basically building towards a final outcome, and the final outcome was this morning.
"I've had a long relationship with Mark and he's clearly disappointed not to get the role.
"No doubt they were close, but I can say clearly that the decision was with the full support of the panel, the recommendation that we gave to the board."
A premiership coach with Port Adelaide in 2004, Williams was seen as the favourite to replace sacked coach Scott Watters while Richardson was seemingly unavailable.
On Tuesday night Williams was one of only three candidates to have been interviewed, alongside Saints assistants Adam Kingsley and Simon McPhee.
Richardson initially declined to be interviewed for the position, but St Kilda was persistent in pursuit of the 48-year-old.
He threw his hat into the ring after speaking with Port Adelaide coach Ken Hinkley and chief executive Keith Thomas.
"I had spoken to St Kilda a couple of weeks earlier and mentioned that it's something that unfortunately I can't do at this stage given my commitment at Port Adelaide," Richardson said.
"Then as things unfolded, on Tuesday night my manager rang and said they're ready to go, in terms of presenting to the board.
"I've had aspirations to coach my own team for 10-12 years now and to finally get that opportunity is something I'm incredibly excited about."
After Watters was sacked on November 1, the Saints set up a six-person committee to identify their next coach, chaired by Pelchen.
Club board members Andrew Thompson and Danni Roche joined recently retired Saint Jason Blake, former Geelong premiership captain Tom Harley and consultant psychologist Matti Clements on the panel.
The panel made their recommendation on Monday morning, with the Saints board ratifying the decision shortly after.
Port Adelaide was reluctant to release Richardson, who held a significant role at the club as director of coaching, meaning both clubs were held in talks for most of Thursday to negotiate his path to Seaford.