Alexander also promised to hold a post-season review which will take place after this coming Sunday's (4.30am AEDT) Cardiff clash against a Wales side under pressure following their last-gasp 33-30 win over Japan - the team that beat the Springboks at last year's World Cup in arguably the greatest upset in rugby history.
Despite that shock loss to the Japanese, South Africa recovered to finish third at the World Cup after being edged out by eventual champions New Zealand in a tense semi-final.
Following the tournament, coach Heyneke Meyer resigned and was replaced by Coetzee, who has since presided over the ensuing slump.
"It's been a tough year and I can take responsibility for a lot of things," Coetzee told a Cardiff news conference.
"But I only got started in April and there are challenges in South African rugby that are unique to us.
"There's no excuse for the poor performance, but there are discussions happening and it is looking really positive.
"This is a great opportunity to address the problems and clean out the wounds. We need to diagnose and treat the symptoms of it.
"In the 21 years since our World Cup win in 1995 we've only had four good years, and if we continue what we've been doing you're not going to see any different result.
"There's a lot of dark clouds, but the silver lining is there I believe."
Former Springbok coach Nick Mallett said that the huge exodus of players in the prime of their careers to wealthier overseas clubs had also played its part in the woeful run of results.
Coetzee has also found himself receiving sympathy from the rest of the rugby world for, unlike any other leading coach, having to implement a "transformation" policy in selection designed to compensate for the all-white nature of South Africa's apartheid-era teams.
But South African rugby journalist Mark Keohane, one of Coetzee's staunchest critics, has pointed out that it was an all-white side that the Springboks lost seven Tests in a row from 1964-65.
For all that Alexander warned against finding "scapegoats," Coetzee accepted that "it's my responsibility to get the team out of that hole."
He added: "Saturday was dreadful and no-one can be happy with that result. We are in a bit of a hole at the moment and the buck stops with me."
As for his own future, the 53-year-old former Stormers coach said: "Whatever happens, happens.
"But I've got a four-year deal and we want to end this tour on a good note."
Later, the Springboks said Eben Etzebeth would miss the Test against Wales after suffering a concussion during the defeat by England at Twickenham two weeks ago.
The lock, who did not feature against Italy, will return home.
"We won't call up a replacement because we currently have sufficient cover in that position with Pieter-Steph du Toit, Lood de Jager and Franco Mostert," said Coetzee.
"There are also some utility forwards who can provide extra cover if needed."
