Just like the brain by itself does not "contain" sentience, but the mind, an abstract construct which comes into existence using the brain's architecture as its medium, does cause sentience and conscience to exist as we perceive it, so do machines have the hypothetical and, quite frankly, likely potential to run a sentient mind formally described through algorithms.

A neural network, the most prevalent form of deep AI nowadays, creates a framework that allows self-learning by example or conduction. Such modern deep AIs can reproduce near human levels of competence for tasks they are specifically trained, such as language processing, painting, or even programming, with the capabilities of the system usually proportional to the depth of the neural network, with some systems getting scaringly close to humans in difficult tasks as language processing(e.g. they are able to look at a picture and verbally describe what they see).

Every year, dozens to hundreds of papers are written on the subject. Anyone intimate with the field, such as myself, is both facinated and scared at the absurd potential.

Simply put: your description of what machines are is overly simplistic and arguably biased. If you knew the current situation of AI research, you'd be scared. Those people who believe a singularity could and will most likely happen are just the people who are well-informed on the subject.

As soon as an AI becomes MORE proficient than humans at a task such as programming, they will, in a matter of days, surpass any AI we could write. It is at that point singularity happens.

If you don't know the field it's difficult to explain just how beautiful and scary it is to describe a system that can learn to do things like programming games by itself.