The Indian state of Kerala has a communist government and India's highest HDI

As far as I know, the princely state of Travancore was not under British rule (https://en.wikipedia.org/wiki/Travancore) and the quality of life, education and health care often rivalled or exceeded many parts of Europe.

The primary, middle school & women's education always figured high in India in contrast to Europe where education was only for the elite until about 150 years ago.

Systematic dismantling of traditional education systems and disenfranchisement of temples which was the support structure of the society was common in territories under direct British control.

https://www.amazon.in/BEAUTIFUL-TREE-DHARAMPAL/dp/8175310952

Documents in detail the uprooting of the Indian education system and substitution of elitist English education.

I think HDI stands for Human Development Index. The cited article mentions neither HDI nor Human Development Index.

Some history - It was the first time in the world (as far I know) that a communist government was democratically elected and came to power without a violent revolution. (Funnily, other communist parties around the world criticised the indian communists for this!). This really rattled the United States of America, and they applied tremendous pressure on the indian government to deal with the "communist threat" in India. Nehru was reluctant, but knew that India couldn't ignore the demands of a world power it needed help from, as India was still struggling to sustain itself as an independent nation. He invoked a federal power that allowed the Government of India to dismiss any local government in a state and take direct control of the region under the pretext of the deteriorating law and order situation in the state (huge anti-communist protests had started in the state by religious organisations, the opposition political party and some claim even the CIA - https://indiatogether.org/kerala50-government).

To Nehru's credit though, he held steadfast to Gandhian ideology and refused to give in to US demands to ban the communist parties in India as long as they swore to abide by the indian constitution and didn't advocate violent politics.

The title seems to attribute the higher HDI solely to the communist government which is misleading. In fact, Kerala has strong anti incumbency and hence the communist government is usually elected every other election cycle. Indian state of West Bengal had an elected communist government for 25+ years but the state didn't fare too well during that time.