I don't think there are any major problems with your proposal. The chips themselves will have documented limitations (specifications) that you would need to be aware of. I think you would find that most manufacturers of washing machines, toasters, TVs etc. do not also manufacture processor chips, so they too would be relying on commercially-available chips such as the ones you mentioned.
This brings us to the IDE. It is basically a "wrapper" around a C++ compiler which is available for anyone to use in any project. The Arduino libraries themselves may have licensing you would need to be aware of, the same as if you used a library not associated with the Arduino.
See this page: Licensing for products based on Arduino:
- You can freely use Arduino hardware and software for your personal purposes, or within your company or organization, with no restrictions. The rules described in this document apply only when you distribute products based on Arduino to third parties or you make them public in any form.
- If you comply with the open-source licenses of Arduino, you can distribute your software/hardware products for free or commercially. This document provides a summary of such rules for your convenience and does not waive the need to get professional legal advice.
I am not a lawyer, and if you are worried about the legal aspects I suggest you get legal advice. However I doubt that a small-scale project would warrant anyone getting their lawyers involved. For one thing, it would be hard to prove that the 0s and 1s on your chip were produced using the Arduino IDE, or any of its libraries.