Difficult cold start

Any ideas how I can check/improve fuel system? Carburetor has no noticeable play in main axle.

This is my plan what I will try after driving season:

1. New spark plugs, wires, points and timing check + valve adjust

2. Add relay to coil power feed to provide better voltage to coil. Another relay for resistor override.

3. New "mini starter" and battery

4. disassembly carb, give it ultrasonic clean and put repair kit to it.

5. ? No idea if those won't work.