As you probably know, Major league Baseball has implemented such a program. It is subject to an appeal by the manager of the team.  it is limited to three or four requests/game though and it goes to New York for a panel to decide.  it can go either way where the call is upheld or overruled.  I don't watch football as a general rule, so I cannot speak for that sport.  To me it would be easier to not have to go to a New York based panel of "judges" to rule on the disputed play. 

Having been a basketball and football referee in a past life, I appreciate your discussion of bad calls.  Even when an official is in the right place, sometimes he or she cannot see what actually happens.  That is why there are now 3 basketball referees where there used to be 2.  More eyes means fewer bad calls.  Even at that, however, since each official has a specific responsibility, there are many times when only one set of eyes is focused on where the specific action is.  Too frequent use of instant replay results in interruptions to the flow of the game and longer games.  There has to be a balance.  And even with instant replay, it often is the case that not enough evidence is available to reverse a call which is different from that call being confirmed.  If multiple camera angles and slow motion cannot generate the evidence needed, imagine the situation for the referee who had to make that call in a fraction of a second from a single vantage point at full speed.  

Your comparison to quality inspection is good.  It has long been recognized that 100% inspection does not translate into 100% accuracy.  In manufacturing and service, we recognize that it is better to improve processes so that 100% inspection is not required to assure quality.  That is a difficult if not impossible approach for sports.  

There is a fundemental flaw in your assumption, and we all know what assume means, don't we. The flaw is that the officials consistently get the call wrong. This is not the case, for almost all the calls are correct and when a call is made in error, it is actually the natural variation of the process. Besides, there are only 2 or maybe 4 calls in any game that might be disputed out of the literally thousands that could be made, given the 7 officials, 22 players on the field for each play and on average 154 plays per game, that would be 23,100 opportunities in a game for a bad call. Hmm, a system with 160 DPMO defects (disputed calls), and the defects aren't life threatening, and you want to fix it?

Besides, even with the instant replay of American football and Baseball, 90% of the disputed on field calls are upheld. So, that 5 minutes of reviewing the 7 different angles and 3 or 4 people with ultra slo-mo is just wasted time and effort.