Back from the tip shop - very uneventful. Except I now have a good working remote for our VCR (yes, we still have it connected to the tv. In my defense, the Betamax is packed away).
Anyway, back to the Echo;
The bit in your fingers in the first photo is indeed the metering diaphragm. It needs to be flexible - if it is rock solid, stiff, or crinkles like a Smiths chip packet when you wobble the middle bit up and down, it is old and aged hardened.
You can soak it in brake fluid for 2-3 days, then wash it off with petrol and reinstall. If it runs, then you have found your problem, but replace it.
I am currently experimenting how long the brake fluid fix actually works for - so far 3 months fine.
Take the 2 screws out near the primer bulb and remove the primer and the carby body under the primer - there will be a small mesh screen (like in the photo) - remove it carefully and shoot carby cleaner or degreaser through it.
With the mixture screws, the benefit of having usable screws is you can easily adjust as required. The downside, is anyone can - often they are buggered with.
Screw them each clockwise until they lightly seat (stress lightly). Count the turns (do it slowly as I always lose count if there is background noise). It should be around 1.25 turns from where they are currently to being lightly seated.
If it is, turn them back out to where they are currently. Then turn the one marked L (right in your second photo) anticlockwise 1/4 turn.
If it isn't, write down the number (just in case) and set each at 1.25 turns out. That should be enough to start it and for it to somewhat run.
It is then a matter of adjusting the low jet so you get a good idle as well as pick up off idle to full throttle - you should be able to mash the throttle and the engine rev up without hesitation. Then adjusting the high end for the right rpm.
My first machine was a whipper snipper - 1982 MSAA Craftsman with a sliding barrel float carby and ignition points, closely followed by a alroh edger (4 stroke tecumseh) and 4-6 weeks later, a 1989 Victa PT.
They can be really fiddly to work on, other times really easy. I once grabbed a Talon Farm Hawk trimmer (that had been used on a farm), off the curb on the way back from New Norcia.
Tore the thing to bits over the next week trying to work out the problem.
Turned out the 2 spark plugs I was trying (1 older champion and a shiny new NGK) were both buggered. It was a fake NGK, and I enjoyed smashing to pieces with bricks and a big hammer

The Talon run like a champ with a good plug.

Regards
Tyler