[RPG] Is the game unbalanced without inspiration

No, Bardic Inspiration can't cause a critical hit

Cutting Words states, "...subtract the number rolled from the creature's roll.", so the number is being applied to actually change what you rolled on the d20.

Bardic Inspiration states, "add the number rolled to one ability check, attack roll, or saving throw", so the number is just another bonus on the attack roll.

The difference in phrasing is very small but important here. Terms like, "attack roll", "saving throw", "ability check", are common phrases in 5e used throughout the book. PHB, p.195, explains that you make an attack roll by rolling a d20 and adding appropriate modifiers. If it weren't for the Jeremy Crawford tweet it would be sensible to assume that the, "roll", in Cutting Words refers to the, "attack roll", but that tweet exists so instead it refers to the actual roll of the d20. Bardic Inspiration specifically says, "attack roll", so we can be relatively certain that it is just another modifier added to the roll.

The spell Bless uses the same phrasing as Bardic Inspiration, "...and add the number rolled to the attack roll or saving throw", and there is no reason to believe that it grants (or bizarrely prevents) critical hits.

Take good notes and remember for the players.

1. Take good notes. Don't rely on your memory for what happened. Instead, take notes during the game and then allot about an hour for going over and adding to those notes. Basically what you want to have when you're done and packing away your notes is a little kit of information that will tell you everything you need to know to keep running the game. Expect to completely forget anything you don't write down, and that will make sure that you have useful notes. Treat it like you're preparing notes for another DM to pick up where you left off!

   Then, weeks or months later when it's time to prepare for the next session, you can take those notes out and read them as the “background” for the next adventure, as if you weren't even there.

2. Remember for the players. During the game, be your players' memories. When you bring in an NPC that they previously interacted with, make that part of their reintroduction. (E.g., “A tough-looking woman with a broadsword on her hip enters the tavern. She's Meredith, the mercenary that you worked with briefly but parted on poor terms with. She hasn't noticed you and walks over to the bar…”)

   By being generous with this information, you make up for what the players are missing: immediate recall of what just happened last session.

These two things are all you really need to make a session work after a long time. Provided you have players who want to be gaming (i.e., enthusiasm waning due to long breaks isn't an additional issue), these practical steps all that's necessary to patch the time gap.