It is probably at the end of its life cycle and new shoots will come up because I see a young pup ready to take over.

I planted mine in the ground and dug them up over the years. Started with one culm (a gift from a friend) and from that one it multiplied to probably 25 or more. Gave several away. Anyway I tried saving the mother plant indoors over winter but that never happened. The new banana plants would come on and grow 10 feet if the culm was big enough. After all this observation I started cutting the entire plant down before digging up the culms. One thing I learned too is do not store the culms in a dry space - they will not survive. Keep them moist for months and plant out again. My last season (2017) I just dropped the culms on the ground and let them root in by themselves. Not as big this way but still six feet or so.

Here was the ones I dropped on the ground in 2017. The temporary fencing is 3 feet tall. These are the ones I stored dry and lost them all.

One of the mother plants I tried saving in 2015

and the grove I had going (this was late October 2015) and I had 3 more stands not in this photo.

This is the recommended way to save the banana culms for the next year (one of the trugs has elephant ears in it). Just dribble a little water in there to keep them moist and happy while dormant.

Anyway I was able to save four babies (I potted these four up and put under lights) the year I stored the big culms in a cardboard box and didn't water at all during winter. I put these 4 babies out the next spring (about six inches high) and they grew about 3 feet.

Here are 3 of them in 2018. I never got around to digging these up for winter and lost these 3. The other one I had in another location did come up from the ground so dug it up a week ago to keep the banana legacy alive. HA!

My advice to you is to cut the dying plant and let the baby/babies take over.

Thanks a lot for your answer! It gave me some idea on the process.

As I leave in an apartment and there will not be any option in the summer to dig it into the ground I'm still thinking how to save the mother tree and I found this: https://www.youtube.com/watch?v=6CnljQdkCEM - what do you think of cutting the trunk somewhere in the middle and letting it grow up from there as I think that for now the new leaves are just "stucked" in some way and they don't really want to grow up.

This is specifically what I want to do (please correct me if anything is super wrong about this approach):

1. Cut the mother trunk in the middle and get read of the old leaves (I've heard that those can be used as fertiliser)
2. Separate and remove the baby plants form the mother plant -> I've no idea how to do that to be honest
3. Plant the baby plants in separate pots and keep both mother and babies alive :D

What are your thoughts on this plan?

The culms continue increasing in size so if you want a big plant then keep your bigger culms. You can easily remove a baby off the mother plant but is much easier if you can see where to divide it (in other words, dig the mother plant out of the pot and shake the loose "dirt" off so you can see where the new plants are. Separate either by gentle pulling at the culm or use a knife or shovel to separate them. Easy - peasy.

Here are two of the four tiny banana plants I potted up in late 2017. This was January, 2018 and I lost the big culms by letting them dry out to dust.

PS, I seriously doubt you will save your mother plant but I am often wrong. Banana plants grow quickly to even play around with trying to save mother plants so I just chop them off close to the ground and dug up the culms for saving over winter. Good luck in your future endeavors!