In this case I have to be honest and say I would leave
Not just because of the cheating but because of the alcoholism mainly, the cheating is a symptom really. But also peoples character doesn't change because of drink they just act how they would if they didn't have inhibitions which I think is worth bearing in mind.
I would also find it v hard to forgive putting my sexual health at risk as I am fastidious about that having reached age of consent at the height of the AIDS crisis.
If he is quitting drinking is he trying to do it on willpower alone? As that very rarely works ime (family full of addicts mainly alcohol)
My own observation of those addicts is that the ones that successfully achieved sobriety
1 completely owned it (the fact you're reluctant to acknowledge his addiction suggests to me he is too)
2 accepted they couldn't do it alone
3 used a sober support organisation/group - aa is the one everyone thinks of but it's not for everyone there are other eg nhs support, church groups
As I thought - you misunderstand what alcoholism is/can be there are a lot of myths that prevail. He needn't be drinking often it's the fact he can't control himself when he does drink is the key factor, it's his relationship with and response to alcohol that is important. There are regular drinking alcoholics, infrequent drinking alcoholics and binge drinking alcoholics which it sounds like he is
That individual counselling didn't work for him in the past doesn't mean it won't now. He's in a different place, then there are factors of motivation and a different therapist etc (he needs to find someone good that he gels with. Bear in mind in uk ANYONE can call themselves a counsellor it's not regulated, psychotherapists and psychologists are)
You have a lot here to consider. If there are children involved in all this I would definitely say leave.