Denied, because I have won Mannhattan without a medic AND Rottenburg without an engineer.
Essential classes, but if good enough players, can be done without.
Not saying you should aim to play MvM without them, but it can be done.
Also, regarding that "Sydney Sleeper" is bad in MvM is also just not true. Jarate is often on a cooldown and while it can be upgraded to recharge faster, the sydney sleeper is the #1 item choice if you want constant jarate on a giant.
Of course it's not too necessary given that you have a scout in your team who will also mark him for death.
Also, the blutsauger is arguably the worst medic weapon to use in MvM. You're going to be healing your teammates, you rarely have a good opportunity to use the only ranged offensive weapon that medic carries. And blutsauger comes with -2 health regenerated per second, so that's making you regen slower for not using it.
Half-Zatoichi doesn't mean you must go full 100% demoknight. While the stats still apply; the 1-hit kill..so does it apply on the robots. Having played a lot of Degroot keep, I find it useful when you want a funny, challenging way to kill the samurai demomen.
For the record, the last time I played Mannhattan we won with our engineer using the gunslinger.
As for Ali Baba's Wee Booties/Bootlegger, that's not a terrible choice.
If a Demoman isn't planning on using his grenade launcher and is going 100% stickybomb launcher, might as well give him +25 hp in which it allows a greater overheal, no?
Regarding your comments on the non-gunslinger engineer, I for one hope for your own sake you don't purchase the mini-sentry upgrade.
It's a waste of 500 credits for a gun that contributes too little to be worth your investment. You rather buy full on crit resistance with those 500.
Two-Way teleport is situational. I never find a good opportunity(aka, don't ever see the need to get it).