We have a glass cooktop.....about 5 years old & the only thing we don't use is the self-clean feature.....even if we didn't have birds, I wouldn't use that thing anyway.....the one time I've ever used one was one too many times for me.....in fact, I turned the breaker off so I could stop the cleaning process...you can't open the door otherwise.....
Since your oven has quit, it's late in the day to send the manufacturer a letter, but you could send them an E-mail similar to this:
We are interested in possibly purchasing one of your products, however we have companion birds in the house and are concerned about products that might contain parts or have been manufactured with or using PTFEs or PFOAs. Please advise which of your ceramic/glass top stove/range with oven products are free of PTFEs and/or PFOAs.
Thank you for assisting in our decision.