I haven't read the whole thread, so something similar has probably been brought up, but I'll say my piece anyway (although I'll leave a key point until the end of anyone is interested).
The main reason I'd be in favour of five substitutions is due to other leagues doing it. They are getting the opportunity to rest players more often throughout the season. This only disadvantages our teams in European competitions, something our FA does historically anyway. Other leagues are more generous in moving fixtures and are doing so with the substitutions. Looks like the English teams will just have to keep doing it the hard way, so to speak.
I largely agree with Jamie Carragher that five substitutions is more for the reverse of what the smaller clubs think it is. Big clubs want it for games where they are perhaps 3 goals to the good with less than an hour played, or extensive leads at half time. Contrary to popular belief, I honestly think that hardly any teams have enough attacking quality to change 4-5 forward and bring on equal or better ones. If Man City need a goal, they can bring on Jesus and Torres to replace Sterling/Aguero/Mahrez... Is that really a big deal? They can do that anyway with three subs allowed. Who else do they change? Foden & Gundogan on for De Bruyne & B.Silva? I don't think being able to make another change or two is a huge deal. We have even seen City & Liverpool making one, or even zero changes in games they aren't winning because their XI will be arguably less likely to score. This just proves the point that five substitutions is there to give players a break in games won, rather than chase a result whilst losing or drawing. You don't often take off better players for worse ones when you need to win, even at a big side with a stronger squad.
My last point, and probably main one at that, is that perhaps there should be some kind of compromise? I have seen people suggest allowing four for the second half of the season which I'd be fine with. However, what about making it either 4-5 subs but 1-2 of them must be home grown players? Or even players under the age of 21?
This gives clubs the chance to rest important players more often, but also give some much needed experience to their younger players.
5 subs with 1 under 21 and 1 home grown minimum would be ideal IMO.