Agree on slowdown.Extra reactions would make things like reach broken, which turn out to be balanced in 5e because of the hard rule of only one reaction per round.
Reactions are cool conceptually, but the interruptions slow down gameplay.
Balancewise, it might be ok to spend the next action on a current extra reaction instead. Or get an extra reaction per short rest, or something like that.
The real balance problem is not getting mulitple AOOs, but rather using multiple defenses and spells in one turn and not paying a opportunity cost for making an opportunity attack.
For example if I have 3 reactions in a turn, my EK can use a reaction to make an AOO, then use a second reaction to cast shield if an enemy hits him and then still have a final reaction for absorb elements if the dragon breathes fire on us.
The current system really makes for a high opportunity cost in using a reaction in a situation like that (which happen often). If you are facing a dragon and his allies you probably are not taking that AOO at all and you are going to debate using shield to keep that absorb elements in your back pocket.
Add in things like silvery barbs, a warlock multiclass and save or suck feats/abilities/spells and ready actions into the mix to be used following an AOO or other reaction and it is going to be very complicated and OP with varied combinations to exploit this seam.