Plants tilt usually because they are turning toward light. I helps to turn pot about a 1/4 every week or two, depeding on light. If you have outside space available, you could keep it outdoors, but make sure it is acclimatized to change, or it may get sunburn.

If most roots have rotted, plant was kept in 'soil' that stayed too wet for too long. While your plant should survive, that siuation should be solved and avoided again. Cactus mix is often not as well draining as it should be. Simple way to improve it is by adding lots of sifted perlite or pumice (whatever you can get). By lots I mean about half of total. Adding just a bit will not help. Excess water should run out of drainage hole. I am not sure why you use layer of activated charcoal at the bottom? Also, what are the big black chunks in your pot?

You did well by trimming damaged roots off. Aloes usually grow new roots quite readily, as long as they do not sit in soil that is too wet. Black dots and soft leaves were signalling the problem, so good you took plant out and repotted. Hopefully, there was no rot set into stem yet. Was it dark-blackish and/or soft?

What rina said.

More light! Can it live outdoors at least for the summer? (Expose to direct sun very gradually, to avoid scorch.)