How do you improve at chess if you don't have time for long games?

I'm trying to push my chess rating past around 1000 and right now, I seem stuck around the same ability level. I've done tactics and puzzles, but I'm not very good at spotting them in game and sometimes I just make simple mistakes. I've been advised many times to play long games well and analyse them (15-30 minutes) but most of the time I either don't have time for long games or don't have the patience to play them.

Is there any other ways to get better at finding tactics and just generally improving or do I have to force myself to allocate time for long games?

You need more time, as there’s no short cut to improvement.

If you find a way to get better without putting in the work, let me know.

You need to build good habits first and it takes time. You need to go through your thought process when choosing a move slowly so you can weigh factors. Over time it will be part of your subconscious, and only then can you play well fast. Ironically, playing a lot of fast games won't help you play fast better. You can't expect to make better decisions with little time. Generally, people who play fast well are people who play slow well and have tons of slow games giving them a lot of experience in a lot of different positions so they have something to compare to more often than not, and so they make better decisions fast.

#1

"sometimes I just make simple mistakes"
++ A rating of 1000 is a sign of frequent blunders. Always check your intended move is no blunder before you play it. That alone will bring you to 1500.

"I've done tactics and puzzles, but I'm not very good at spotting them in game"
++ How much time do you take to solve a puzzle? How much time do you take to move in a game? Time per move in a game should be longer, as you do not know if there is a tactic or not, so you have to always assume there is one.

"I don't have time for long games"
++ Better 1 long game than several fast games. 15+10 is the best time control.
"I don't have the patience to play them."
++ Chess requires patience. It is a complicated game. You cannot play well unless you think.

You can always play daily games. That way you can take as long as want in a position, and if you’re called away it’s no big deal. However, you need to develop the patience to sit and play for an extended period of time if you really want to improve. Adjust your schedule a bit if need be. I play one 2+ hour game a week. Any other faster games are just for fun and to relax after long workdays.

Solving puzzles helps you learn to spot combinations in positions, and slowly will build pattern recognition. But thought process is the most beneficial which takes time- you have to be able to look the board over and assess what’s going on and then come up with a good move based on that. Have a great chess adventure!

You will not improve if you are not willing to be patient

You can't. There is no escaping that chess is time consuming and the only way to improve significantly is by playing longer games.

100% true.