{Sigh}
Where do I begin?
1) The circuit isn't mine, it was a modified version of one provided by the OP, GeirA. If you scroll back up, you'll see that GeirA was playing with said circuit, and had a problem/question with it, which we hoped was answered.
2) I provided this new circuit after efron pointed out some problems, to help out GeirA, as folks on this forum are want to do. That's how we roll.
3) Efron then pointed out that I had made a couple mistakes in my modification, since corrected. I may plan on running all my schematics by efron in the future, as he's got the eyes of a hawk.
4) We are now just waiting to hear back from GeirA as to what he found wrong, and how it was corrected. At no point was it asked if there was a better solution than two transistors, two caps, four resistors, and two LEDs (ten parts) - it was only asked what may be going wrong with the circuit provided.
5) Your suggestion isn't what anyone here would call "easier". The parts count is the same (10) - but the 555 will only operate from 3-15V, whereas the transistor circuit will easily run from 1.5-40V using off the shelf components. Both circuits will need to be tweaked for desired function. The footprint/layout will be about the same (two TO-92s can easily fit in a DIP-8). There is no advantage of the 555 over the transistors in this case.
6) GeirA mentions that he is a beginner, and doesn't have many parts stocked (the reason he's using 10uF instead of 1uF). A 555 circuit won't help him one damn bit if he doesn't have any 555s on hand. Also, it won't help GeirA with the R/C time calculations, as the 555 operates differently with respect to timing (1/3-2/3 voltage rails on the tank circuit for the comparators).
Lastly, Electroalek, when it comes to electronics, beginners should take baby steps. GeirA is learning transistors, caps, LEDs, and our friend the R/C time constant - just like he should - before tackling the harder stuff. Your suggestion would be the equivalent of someone posting a question about how a 2-stroke engine works, and someone like you tells him to abandon the 2-stroker and go right on to a Rolls-Royce V12. You're forgetting that there is a lot to learn about that 2-stroke that'll be handy to know when tackling that V12.
Baby steps.
nop