I agree that Chinese companies are usually targeting China first before taking on the rest of the world, while American companies usually have a more global mindset to begin. However, I think you are quite incorrect in thinking there are "few" companies with a global footprint, it is much larger than you realize.
There are many Chinese companies with a global footprint, but they are not consumer facing, for example you have Fuyao automotive glass with manufacturing in China, US, and Germany, LK Industrial is supplying the Gigapress to Tesla (Directly in China, through subsidiary Idra SpA in USA), of course Huawei networking.
In other cases you have branding or OEM deals, ie. GE Appliances is a brand of Haier (also huge global footprint), or HikVision which sells a great deal of their goods to resellers.
Basically only recently did Chinese companies have fully professional management. 20 years ago, it's still common to find executives without formal education, or working up from the factory floor. They simply lacked the global experience to create global success. Often they would venture outside of China and face "culture shock". Acquisitions often failed due to management clashes. Chinese managers are now much more well-travelled, Chinese education has more global focus (ie. professors from all over the world), and executives might have foreign education credentials. You see DJI and TikTok quickly became leaders in the field.
I would add Korea is okay, but they have very few brands owing to their Chaebol economic policy where basically everything is owned by LG, Samsung, or Hyundai groups.
