The intake valve is open all the time whether hot or cold, but it only affects starting when cold. I can only guess why. Maybe it has something to do with the propensity of the gas to ignite. The screw above the nut is what's used up. Eventually you'll have to take the screw all the way out. So however many threads are below the nut is how much you have left. Switching cams wouldn't change anything because it's the part of the cam with no lobes that would affect it. Now I see why they sold it cheap. Things are starting to make sense. Luckily it won't take much to fix it.

26 is a little small but it might work. It would be easy to tune by having plenty of vacuum. I originally designed mine for a 32mm, but the 32 I used was designed for a 2-stroke and was impossible to tune. I finally decided I didn't have a big enough engine to produce the vacuum required to get a 32 tuned right so I stepped down to a 30. I really didn't care about performance anymore and just wanted it to work. I'm of an opinion now that smaller carbs are better. My Honda 250 (229cc) has a teeny weeny 22 or 24 or something ridiculously small, but doesn't seem to have problems revving. That is the most reliable quad I have. Always starts in any weather (by virtue of the strong vacuum on the carb). It's a pushrod engine with no oil filter. A 28 is probably about right for you. I'm overbored 1mm with the 250 cam with some exhaust mods so maybe that justifies a 30. Just keep in mind the bigger the carb the harder it will be to tune.