Point 1:
Some of are old enough to remember the days before substitutions were allowed. If a team had two players taken off the pitch through injury in the first ten minutes of the game they then had to play the rest of the match with 9 players.
Substitutions were brought in SOLELY to allow for injured players to be replaced. However, rather than imposing an [impractical] test upon the level of injury, team managers were allowed to bring on substitutes for any reason whatsoever (up to the maximum number allowed which, in most competitions, was usually two).
So (based upon the original reason for allowing substitutions) managers are already allowed to replace three INJURED players; if they choose to replace players who've not suffered actually injuries they have no right to complain when they later run out of substitutes.
Point 2:
Many people genuinely suffer from back injuries which cause them excruciating pain but which doctors can't find any physical cause for (or which take many months of tests to diagnose). Equally, many people 'throw a sickie' at work by claiming (non-existent) back pain. How could a doctor possibly tell, within only a minute or so, whether a player's 'back strain' was genuine or not?
Similarly, when refereeing local league football matches, I've had several situations where (following a clash of heads) a player has required medical attention because he hadn't got a clue where he was. (i.e. severe concussion). It wouldn't be hard to fake such a condition.
Most GPs would be happy with a 50% success rate on correctly diagnosing a medical condition upon a first consultation. (The statistics show that the success rate is actually considerably lower than that, unless you accept "it's probably a virus" as a diagnosis!). Doctors have a hard enough job in diagnosing a problem when the patient is almost certainly being truthful; it would inevitably be much harder if there was a possibility of deception.