Pictures of the auto-choke setup on the carburetor?

There are some schematics, but nothing specific in the Videos tab

https://www.google.com/search?q=96+hyundai+1.3+carburetor+schematic+diagram&rlz=1C1QJDB_enCA643CA644&sxsrf=AOaemvKagK0Xk_9THvl03TBT3bjV43uT7g:1640470987626&source=lnms&tbm=isch&sa=X&ved=2ahUKEwjq2uzT_v_0AhWWQs0KHcZFBWwQ_AUoAXoECAEQAw&biw=1920&bih=880&dpr=1#imgrc=HM4JEUeXUH8DNM

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At about the 35minute mark in this hour + video there are pictures of your carb setup.
Maybe it will help -
Auto choke for Honda EU3000iS by Pinellas Power Products

Possibly could be a throttle position sensor, outside of that I would recommend replacing the choke cable, and maybe the choke itself.

McCulloch u spray starter fluid Power Generators Questions ...

www.fixya.com > Forum > Tags > u spray starter fluid > Power Generators

McCulloch power generators u spray starter fluid related questions and ... Question about 13HP 338 Cc Portable Generator # FG6000MK ... My 5700 watt 11 HPMcCullough generator wont ... My generator will start if I spray it with either but won'tstay running in less I keep ... The bowl has gas in it and the needle is clean.

This generator McCulloch fg6000mk 11hp has no spark. I've ...

www.fixya.com/.../t7785814-generator_mcculloch_fg6000mk_11hp_no

Jan 13, 2011 - this generator McCulloch fg6000mk 11hp has no spark. I've checked the gas bowl , a small amount of water in the - McCulloch ... water in the bowl not much also sprayed a small amount of starting fluid in .... My 5700 watt 11 HPMcCullough generator wont start. I'm getting good gas to the plug but no spark.

Won't start. I have a never used McCulloch 5700 11HP ...

www.dqrkc.com/support/t1325485-start

Dec 2, 2008 - Question about McCulloch 13HP 338 Cc Portable Generator # ...evaporated. you might tryt a little spray starter fluid in the spark plug ... My generatorwill start if I spray it with either but won't stay running in ... The bowl has gas in it and the needle is clean. ... My 5700 watt 11 HP McCullough generator wont.

1325485 Won't start. I have a never used McCulloch 5700

Have to an ultrasinic cleaning of the carburetor and see if that will clean up the problem,

Bad on parts though most are NLA. but a carburetor from a Honda 11hp horizonal might work on this Chinese Honda cloned engine.

http://www.mcculloch.com/ddoc/MCCO/MCCO2008_USen/MCCO2008_USen_FG6000MK-7096FG6024_966991401.pdf

page 27 or 28 I believe. illustration 10-1

A common problem is clogged jets in your carburetor. Choke bypasses the jets. Clogging due to old fuel happens much faster when there is ethanol in the fuel. Solution: Disassemble and clean out the carbs. If you're lucky you can try putting some sea foam (available at Walmart or auto store) and run it through the engine to unclog the carburetor without disassembley.

I did not understand how you turn on generator, as it is always turned on.
If you use the ON/OFF switch it is stopping the prime mover.
If you reset the AC breaker without load, and it stops - it means that you have short circuit after the breaker. Check the output receptacles.
Try this link:
http://weborder.husqvarna.com/order_static/doc/MOUS/MOUS2008/MOUS2008-FG6000MK-7096FG6024-966991401.pdf 

Sounds like your choke is not closed correctly when you first start the genset. Remove the air filter and look at the throttle plate. It should be closed with a hair line gap. With your finger you should feel very little play on the throttle plate. As you open the choke, you should be able to open it all the way once the engine has reached a hotter temperature. If the engine is still cold and your opening the choke all the way, it will try to die out.

it sounds like the govener screw needs to be adjusted by turning it clockwise and then you may have to tweek the throttle screw. hopefully that will take care of it.