There are three routes, and for your use case — short text, several times a day — the third is by far the best.
Feed the note through an ordinary printer. Stick the note onto a carrier sheet of normal paper, print with the note's position set in the document, and peel it off. This works and it is fiddly, and there is a real risk of the note detaching inside the printer, which is exactly the ending you predicted.
Buy printable sticky sheets. Full-size sheets with a repositionable adhesive backing, made for ordinary printers. You print several notes at once and cut them apart. Reliable, and it means preparing a sheet at a time rather than printing on demand.
Use a small thermal label printer. This is the answer. They are cheap, they take rolls of small labels including repositionable adhesive ones, they need no ink at all, and they print a short list in about two seconds from a phone or computer.