Why aren't perpetual bonds more common? Most government bonds and corporate bonds have a maturity date when the principal must be repaid. 

While the few percents of interest every year is generally not a big problem to pay out, when a bond matures the issuer must pay back all the principal at once which can be a huge expense and the issuer can go default if they don't have all the cash at hand by that time.

If the bond issuer issues a bond without maturity, then they only need to deal with the interest payments, which is a fixed amount of money and as the inflation gradually devalues the money it's become gradually less problem to pay it out. 

If the price of the bond decreases or after a good year, the bond issuer may repurchase some of his own bonds from the market on his own discretion to further reduce the interest burden.

From the investor side this is kind of security can be bought from the market and sold when they need the cash just like any other security regardless of the lack of maturity. This way this security can be like a stock with a predetermined dividend.
Since the issuer only need to pay interest, the credit risk is lower, but interest risk is high. 

So far it looks like a perpetual bond is just another kind of security with interesting properties and risks.

But most bonds do have maturity, so what are the caveats? What's the reason that most bonds do have maturity and perpetual ones are uncommon?
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.