Yes, Chile's HDI is quite high for a Latin American country. The problem is, in the same city (Santiago) you've got communes with HDIs above Norway's (Vitacura, Las Condes, and Providencia got HDIs above 0.953) and then there are communes less than an hour away that got an HDI lower than Panama's (San Ramón HDI is 0.786) [2][3][4].
By OECD's data, Chile is the third country with the highest income inequality amongst OECD members [5].
[1] http://documents.worldbank.org/curated/en/496131468228282235...
[2] Calculated with INE's data (2017) https://imgur.com/GKMcGw9
[3] http://reportescomunales.bcn.cl/2012/index.php/Categor%C3%AD...
[4] http://www.desarrollosocialyfamilia.gob.cl/informacion-socia...
But besides that, these trade offs are for Chile’s voters to resolve. Those voters decided that sticking with markets, even in the face of higher inequality, was the way to go.
Anyways, yes, that's something that voters should resolve. Sadly, there's a big discontent towards the political class in Chile (against both left and right-wing) and last elections had low participation (around 50%).
[1] http://documentos.bancomundial.org/curated/es/24225147670682...
Unsurprisingly, many people don’t want to rock that boat. The current center-right government was elected in 2017 by a decisive majority. (The President won 55-45 in the run off election).
The protests in Chile are largely socialists and labor movement that are trying to get policies passed they couldn’t get through the democratic process.