No. They do not.
When I tried to use LLMs to create Zabbix templates to monitor network devices, LLMs were utterly useless and made things up all the time. The illusion of thinking lasts only as long as you stay on the happy path of major languages like C, JS or Python.
It requires as much thinking as it did for me to copy-paste code I did not understand from Stackoverflow to make a program 15 years ago. The program worked, just about. Similarly you can generate endless love sonnets with just blindly putting words into a form.
For some reason we naturally anthropomorphise machines without thinking it for a second. But your toaster is still not in love with you.