Did your doctor give you no indication of whether the conservative (non-surgery) approach was realistic? The inner two thirds of the meniscus is termed the “white zone” because of it’s lack of strong blood supply, and tears to this area generally are not repairable without some sort of concurrent bleeding damage (like also getting your ACL reconstructed). Whether or not your tear occurs in a portion with good blood supply will determine if it has the potential to heal. 

Though whether surgery is necessary or not may depend on the severity of the tear, and whether the symptoms it causes are debilitating (they seem to be messing up your training pretty bad, at the very least).

The nice thing is that the surgery is pretty non-invasive. They don’t need to open you up, so the recovery time is actually pretty good. I had a partial meniscectomy of my medial meniscus a couple years ago, and I was able to walk reasonably well within a week or two, and make a slow return to activity at like 5-6 weeks. It did take a while for me to achieve full range of motion in my knee again, but now that sucker is stronger than ever, and requires only a little bit of extra upkeep to keep healthy.

I know surgery is a pain, but I waited forever to get under the scope, and that whole time my knee would randomly fetch up a few times per week and leave me limping. If your doctor and physio think it’s realistic, try the conservative approach. Be cool on that thing for a while, give the initial injury time to chill, and see just how much it actually affects you.