In chapter 20 of Player Piano, the Shah gets a haircut. The barber tells a story about another barber who invents a machine to give haircuts, thus putting himself out of a job. This is a theme in the novel: note that in chapter 8, engineer Bud Calhoun also invents a machine that can do his job and is fired. We might also relate this to the broader science-fiction theme of the ″irresponsibility of science and technology.″ By building this theme into his novel, what do you think Vonnegut is saying about technological invention and engineering?

Player Piano is centered around themes of industrialization and automation. Vonnegut depicts a world in which nearly all jobs have been automated, and there is an increasing class divide between the small elite class that mostly manages the machines and everyone else, no longer needed as workers.
The protagonist, Paul Proteus, is an especially interesting character. It is largely in him that Vonnegut draws out the tensions of attitudes towards work and progress. Paul Proteus is near the center of the push for automation, while also holding onto an appreciation for hard work that automation is increasingly rendering obsolete.
The two anecdotes you give (of the engineer and the barber) both reveal contradictions in the logic capitalist society has towards workers. In theory, such inventions make it far easier to complete the jobs assigned to these workers, and it would seem appropriate that they be rewards for making such inventions. A logical reward might be higher pay, or the ability for them and their fellow workers to continue to be paid while doing less work, or having more ability to choose what kind of worth they do, since machines could handle the bulk of it. However, this is not the case. The advancement of technology fails to make their lives easier and simply pushes them into a position of lesser social power.
This suggests that the society is not organized around doing the work that is necessary to be done or around rewarding people for their contributions to society. Rather, Vonnegut suggests that workers only receive what little they do because they are needed, and that when they become unnecessary, they will be pushed into a position of even lesser social power. This and the depiction of the insurgents in Play Piano suggest that the answer lies not in working harder or in technological inventions, but in finding ways to directly wield social power. This is a picture of an antagonistic society existing under the veneer of a value system that claims hard work and ingenuity will be rewarded.
A separate theme worth noting is the appreciation for creativity, art, and work that is not simply about producing a product, but about the process. We can think about Proteus's appreciation for gardening with basic tools, or of barber shops not simply as a place to accomplish a task (cutting people's hair), but as a social space and a place for customers and barbers to work together to create hairstyles in a way that has room for artistic approaches.