Hi Adams,
If i can give you my opinion, I would say yes that is enough to be really fit, usually...but if you can afford a guide (really expensive i know...but how much does a life cost...?:=)), even better of course.
Climbing Mont blanc can be easy or become very problematic depending on weather conditions. It often snows in chamonix in august above 3500meters...If the snow turns to ice with low temperatures, then it becomes important of course to be comfortable with cramps (crampons)...and a pick...your progression will be more risky. if it is too warm, not cold enough, it can be more tiring and people not fit enough dont make it. Only 30 % roughly of people who try to climb Mt blanc make it to the top.
It also besomes tricky when it is windy as you know so the most important thing is to check again and again the weather forecast the days before. do not hesitate to ask for advice at la maison de la montagne or even at PGHM in chamonix...At the tourist office, you might get "wrong" (not professional) advice as it is not their job in the first place..but people often ask them...
I climbed mont blanc 4 times, alone or with friends, no guide... (different routes : via aiguille du midi and refuge du goûter) and lived for 3 years in chamonix. Some very sporty people living in chamonix never made it, despite thei being very fit simply because their body cant take altitude. Up to 4000 meters, most people are ok, above, it is impossible (or dangerous) for some people to continue.
the best way to know that is to try !:=) but if you climbed before, you know it...
In my opinion, the best time is at the end of september/october before the trains/cable cars close in les houches (village). Summer time, it is (too) crowded, therefore less pleasant and more dangerous (stones falling, slow progression etc)
I have always avoided spring time (may, june) too risky in my opinion (warm ...snow falling : "seracs")
Do not hesitate to ask if you wish to. we can also meet for coffee and have a look at the maps. I encourage you to go, it is not technically speaking difficult (normal route). if you are fit, motivated (exhausting indeed) and the weather is nice, it is just ..fun!! :=)
Cheers,
Chris