The last case I can think of where a team successfully played five bowlers with the all rounder at 6 or 7 was Flintoff (which was quite a while ago now).
You could argue the Saffers do if you count Kallis as an all-rounder, I consider him a batsman who is also a very good bowler part-time.
Not sure the Flintoff era success was that great, for the narrow Ashes win in which he, Jones, Hoggard and Harmison bowled brilliantly and Flintoff batted well for one of his rare big scoring series you also have the whitewash down under.
We had two good series in a row, I think beyond that our record with five bowlers including Flintoff was ok but not that fantastic. We played the windies and kiwis a lot, he missed the home win over Pakistan but not the loss in Pakistan and draw in India
Series results where Flintoff bowled 80+ overs
01/02 India (a) L0-1
2002 Sri Lanka (h) W2-0
2002 India (h) D1-1
2003 South Africa (h) D2-2
03/04 Sri Lanka (a) L0-1
04/05 South Africa (a) W2-1
2005 Australia (h) W2-1
05/06 Pakistan (a) L0-2
05/06 India (a) D1-1
2006 Sri Lanka (h) D1-1
06/07 Australia (a) L0-5
2008 South Africa (h) L1-2
08/09 India (a) L0-1
2009 Australia (h) W2-1
01/02 New Zealand (a) D1-1
03/04 West Indies (a) W3-0
2004 New Zealand (h) W3-0
2004 West Indies (h) W4-0
I've split it into tough-ish series and not so tough, Sri Lanka were below par in 2002 and the aussies in 2009 but I've classed them as tough(er). That's just four series wins against anyone other than West Indies or New Zealand where Flintoff has played a reasonably active bowling part (80+ overs), that out of 14 series against the better Test sides. Considering we played a blinder in that South Africa series, and played very well in 2005 only narrowly winning the two Tests we did win (2 runs and 3 wickets)
So I'm not sure the evidence is great for the five bowler "Flintoff era". Only once did we beat decent opposition by a two Test margin and that was a below par Sri Lanka at home. We didn't beat India in a series once in four attempts, and arguably our biggest and most convincing win of that time period was when he wasn't in the side and we played four bowlers! (vs Pakistan 2006)
Those two series wins in 04/05 and 2005 were massive, but perhaps not reflective of the state of our cricket which was reliant on individuals to shine, I mean Flintoff had a cracking two series with 47 wickets and 669 runs across the two, but those were by far his best two wicket taking series with the next best just 14 (vs West Indies) and then 13 in Pakistan (L0-2)
For all his talent, I'm not sure Flintoff lived up to it. Maybe he was batted one position too high, although his average at #7 is practically identical to that at #6. For his supposed greatness he had a disappointing bowling average of 32.79 , never taking a 10-fer and only taking 3 5-fers. And only five career hundreds in 130 innings was also disappointing, considering he made a fairly respectable 26 fifties at 1/5