Nginx is a front end cache system , its not a full fledged web server like apache or litespeed. Nginx have some cons, like we cannot have rewrites enabled in easy manner. You either have to link nignix configuration with php5-fpm or you have to keep niginx in front end of apache.
Also it should be noted that nginx cannot be directly implemented on control panels like CPanel / WHM. Yet there is something called
https://engintron.com/
But most hosts does not use engintron. On the other hand litespeed is quite a good alternative to apache and gets the best of nignix and apache without compromising on CPanel /WHM. That is why performance web hosting providers like us, use LiteSpeed+CloudLinux+SSD for best possible performance.
I do not know, if some providers using litespeed for their own site and does not offer for customers, may be they are doing so. But we offer complete litespeed hosting for customers, if you are interested check the shared hosting offers section for more details.
Litespeed is good, if you want to see the performance improvement and coming from apache. We also have http2 enabled. So you will get all the best stack.