I would agree that minimizing investment before you know how interested a child is in an instrument is wise, but that acquiring really solid foundational technique will require a decent-quality instrument. Used uprights purchased from estate sales certainly can be sketchy, but you can get some very good deals on respectable instruments from piano technicians who rebuild old instruments.
I would also agree that having a technically skilled instructor is very important. You want someone who understands both musical technique and the body mechanics involved in playing the instrument, to avoid injury down the road. Any technique that creates pain, tension, or discomfort anywhere in your body (not just in your hands) is questionable. (Other than the kind of surface pain in the pads of the fingers that beginners sometimes experience from unaccustomed contact with the strings or striking the keys too many times while one is still developing calluses.) This includes shoulder, back, hip, and neck pain.
If you listen to and watch the students of a good teacher, you should be able to hear and see patterns that demonstrate the teacher's instruction--internal rhythm, voicing, phrasing, clear dynamics, hand positions (a shared characteristic "look" to the students' hands and body positioning while they are playing), a sense that more advanced students are actually more musically and technically skilled, and not just playing more notes.
It probably won't hurt to start off with the known quantity for a piano teacher, but while you are trying out whether your child really enjoys the piano, you might try attending some student recitals (May and June are good times for this) of area teachers, and getting a feel for the kind of students they are turning out, not only from a musical and technical perspective, but in terms of their positivity, warmth, relationship with students, ability to inspire joy in music in children, etc. The amount of time and effort to be expended depending, of course, on the desire of your child to learn the instrument.
