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This is a great point. To add to the 'stabiliser muscles' bit, I love chaos training for bench. It's fun and completely different to anything else you'll do with a barbell. Attach some plates or kettle bells to the bar using folded resistance bands to increase instability and force those smaller muscles to work overtime. Bar will wobble like crazy. I like doing it on an incline to put more emphasis on shoulder area and increase range of motion. I tend to keep the weight around 30% of my 1rm (definitely start light) and usually on either an off-day or the last compound lift of a workout.
Here's some guys doing another form of it:
https://youtu.be/xerAxVjEgYI?si=XZF3rrb6vKnpQXzZ
Do you film your sets?
At least if you're struggling with bench I would suggest it. You may notice what you're doing wrong. Alongside this, I'd suggest posting a form check with the video.
Make sure you're retracting and depressing your shoulders properly and creating a really stable base for the bench press.
I would personally advise against benching on a smith machine. The bench is the only lift of the main 5 you're doing where the bar should not move in a vertical line, it should move in a slight diagonal path. The smith machine will force you into a vertical bar path, this will not help you. It will also take pressure OFF of your stabiliser muscles, which are the exact muscles you need to improve if you're shaking / wobbling.