Fazekas,
You can use AEBS with an external HDD as Time Machine.
Trust me, anybody is doing that.
That's why people believe Apple is doing a wrong pricing strategy.
If you don't trust me or anybody else, just go to Apple Store, and they'll tell you the exact same story.
AEBS + External HDD --> Time Machine capability.
To minimize your issue, mount the external HDD to your Mac first, run the 1st Time Machine to create the backup, this way is faster.
After finished, unmount the external HDD, and mount it in AirPort Extreme Base Station, AirPort Utility, AirPort Disk something.
And that's it!
That's how I did it. For the past 16 months.