Generally cheap machines have compromises which mean more work for you - less reliable, less accurate, weaker laser, worse software - all of that means a lot more trial and error, a lot more burnt material and a lot more stuff you just can't do.
If you want to laser cut thicker materials, but only occasionally, you'd probably find it much more cost effective (and low effort) to either outsource or find a makerspace which has a decent machine. Laser cutting is quite easy to outsource as it is very easy to fully specify what you want (2D drawing + material specification is all you need).
Generally speaking, for cutting anything above paper or balsa wood, you will need module that costs around $500, higher power modules need decent power supply.
There are some laser cutters from china that claim to have such power for less, but there is not jet "consensus" on any of DYI forums/reddit's that any of them are good jet.
As others have suggested, jobbing it out or joining a makerspace is often the best choice for occasional use.
Had a quick look and the cheapest laser cutter appears to be around $400. The next one is $1500 and then the price just keeps on climbing.
Any first-hand experiences with laser cutters? An option suitable for an (occasional) hobbyist use, but not a total junk either.