Why aren't perpetual bonds more common?
The most famous perpetual bonds are UK Government Bonds known as consols. They weren't issued to avoid the rollover risk you highlight. Rather, their key benefit was liquidity. They could sell new consuls on the same terms as the old consuls and they have enhanced liquidity because it made the new and old issue more liquid. 
They turned out to be a pretty horrible investment in real terms for early investors. Inflation ate away most of their value. 
Despite their seemingly infinite life, because they pay out coupons and we discount the future geometrically their interest rate sensitivity behavior is similar to existing long dated bonds. 
Unless the bond is callable, you generally can't make money by repurchasing your own bond at market prices. If interest rates drop, making you want to replace old bonds, the price of the old bonds goes up until the yield of old and new bonds are equated. Things are a bit different with changes in credit risk but even in that case it can be tricky. 
The product you describe also sounds a lot like preferred stock. The Wikipedia article on perpetuities gives consuls and preferred stock as two examples.