It might not help with the original problem but I use a piece of dowel, that is just under the width of the hub and spacers, to hold those pesky spacers in place, it knocks back out when you put the axle back in. The wheel also goes in easier if you roll it in from the rear on a plank of the right height. As said make sure the calliper mount is seated properly and the chain is out of the way, but still the right side of the sprocket.

Still could not do it. I'm going to wait for my uncle to come by this week, he is a car mechanic but also used to own a MT-03 which he rebuilded as a hobby.

On the following video you can see from a 660 tenere (same sort of rubbers) that the person is able to stomp it tight, but can't do that on my MT-03. It still leaves a gap. I have tried giving it CPR massages, hitting it with a rubber mallet,...

https://www.youtube.com/watch?v=ro8ldj4gQV4

This is the farthest I can get my rear sprocket assembly in:
https://imgur.com/a/jG5OElR

what year is your MT 03?

A 2016 MTO3 set of rear rubbers is Yamaha part number 1WD-F5364-00, earlier MT03's had the same below part number as a Tenere and so many other Yamahas which have the same rear rubbers such as models including things like the 250 V-twin Virago and even the old XT500 (from memory )which is part number 5Y1-25364-00

Where did you get your new cush drive rubbers from and are they confirmed the correct part number?

Brappedout, your picture of the sprocket hub fitted to the wheel hub shows the problem, the sprocket hub should be flush or just under the edge of the wheel hub, I'd suggest you have the wrong cush drive rubbers, they are too fat, put the old ones back in. Don't try and make that fit and ride the bike the chain will come off the sprocket and do much damage, no ones wants that to happen. have you checked the sprocket hub bearing, is it seated correctly?

The old rubbers are the same as the new ones. I put the old ones back ones back in and it was the same result.. something must be wrong with the bearings I guess.. How could I identify a bad bearing?

https://imgur.com/a/FAUgMuh can someone "spot" a bad bearing on this? I know its a crude way but I dont know anything about bearings...

The bearing looks seated correctly. Put a smear of Vaseline, or general purpose grease, or even olive oil on all faces of the rubbers and using the; lube-your-rubbers-for-a-nice-slip-in, principle your sprocket carrier should push straight back in to place. Rubbers are sometimes 'dry' and we all know how that hurts...

hummm....suggest you take the rubbers out and drop the sprockett hub back in, I am assuming it will fit properly. Are you putting the cush drive rubbers in correctly and all the same way. Take a picture of the sprockett hub, the drive lugs side and the wheel hub and one with the cush drive rubber fitted and post them up

ITS FIXED!

My uncle came over and inspected the wheel. He saw that a bearing came out too far, we used some penetrating oil and popped it back in. After that, the wheel was back under the bik in under 5 minutes....

I'd like to thank you all for the replies and help!

FOR FUTURE MT-03 OWNERS STRUGGLING WITH THE SAME PROBLEM:

If you want to put your wheel back after taking it off/changed the cush drive rubbers but cant because of a lack of space, check this bearing:

https://imgur.com/a/lTRWCy2

It should be all the way down. If not, use some penetrating oil and lightly tap it with a mallet,... it will make a 'cling' sound when it is.