Of course the fact that Liang Qichao said it doesn't make it not racist. In fact, Liang bless his heart was pretty racist at that point in his journey. Liang & Yan Fu, who also used the phrase if memory serves, were strongly influenced by Social Darwinism for ex.
'Sickness' is a major trope of modern Chinese history--used & abused by imperialist aggressors & nationalist reformers alike. Prior to 'sick man of Asia,' racists in the US promoted 'yellow peril' discourse, linked to mob violence on Chinatowns & the Chinese Exclusion Act of 1882
From Yan Fu & Liang Qichao onwards thru Mao Zedong [who wrote about the need for Chinese to become physically fit and healthy] & Chiang Kai-shek [whose New Life movement equated good hygiene w modernity], overcoming sickness/backwardness to become strong/powerful is omnipresent
Maybe the most profound exploration of the theme is Lu Xun's short story 'Medicine'. There are no easy answers in Lu Xun. His sword cuts through the callous sophistry of imperialism/racism & the chauvinistic vanity of nationalism/xenophobia.
