Take a look at the ignition system first.
Check the distributor. The shaft needs to have very little side play. If the shaft moves, the points will not stay set. The gap can change under varying RPM, effect the timing, spark quality, and point life.
Check the centrifugal advance. The rotor should turn a few degrees CCW, then spring back when released. If it's frozen, sluggish to return, of moves without springing back, the distributor will need to be removed, disassembled, lubed, repaired... Or replaced with a rebuilt.
Also check the spark quality at the end of each plug wire. Should jump a minimum of 1/4" to ground.
This is important, the symptoms you describe can be a result of weak spark. You mention backfire, exhaust backfire is an indication of intermittent spark. Poor performance and carb backfire can be caused by retarded ignition timing (worn distributor, frozen advance).
Have the valves been set lately? They are probably OK being it idles well, but if noisy, or haven't been set in a while, might put them on the to do list. Be a good time to clean out the gas tank too.
On to the fuel system...
Has the air filter been serviced? Including the wire mesh inside the canister? It is often overlooked. It gets packed with dirt and has to be removed and cleaned or replaced. Not an easy job but necessary. It can also be restricted with mud dobber nests or mous nests. A trial run with the cleaner hose removed will tell the story.
The fuel supply. How is the tank? If it hasn't been cleaned or replaced, good chance it is getting grungy. The fuel flow test, try it again, this time get a clean glass ready, remove the carb drain plug, catch the flow. It will have a full flow at first, but let it continue to run. As the bowl empties out, the flow will slow, as only what is coming through the needle valve will now be flowing. That is the real test, you want to see a continued trickle, not slow to a drip or completely stop. If the flow is insufficient, try loosening the gas cap. If opening the cap doesn't help, start tracing the restriction back toward the tank. There should be a screen in the carb inlet elbow, a screen in the sediment bowl, and a screen inside the tank.
Look at what was caught. If the gas if cloudy or water on the bottom, or rust and grit in the glass, the same will be in the tank and the carb bowl. The tank will need to be removed, claened, or replaced if heavily rusted.
Once the fuel supply is good, the ignition system is right, everything else is up to par, then the carb can be adjusted. The carb will only work as well as the rest of the engine. A sick engine will not operate properly on any carburetor!
All adjustments need to be made with the engine up to operating temp with a thermostat in place.
Sounds like you already have the idle adjusted. Just an FYI, the idle screw adjusts air flow, not fuel flow. Turning it in richens the mix, out leans it. The adjustment is minimal. The RPM must be down around 400-500 RPM or it will not be fully on the idle circuit.
To adjust the main jet, have the engine at idle speed. Turn the main in (lean) 1/4 turn. Open the throttle fully and suddenly, observe the response. If the engine revs without hesitation, turn the screw in another 1/4 turn, repeat the test. Keep doing so until the engine stumbles, then start backing the screw out 1/4 turn at a time. The goal is for crisp response without hesitation when the throttle is suddenly opened. A single puff of black smoke is acceptable.