Yes, but if they were officially allied, China then has to worry about potential F-22 or F-35 took off from some Indian airports and move more troops to the Tibet plateau.All India doesn't have is a US base on its territory, in every other aspect its as good as an official US ally. Even in Ladakh, the outgoing US ambassador admitted that US was giving them real time intelligence . India has also signed all foundational military agreements with US. The notion that India shouldn't be pushed to the west is redundant as its already in their lap.
Unlike now China could mobilize an extra 100k+ Indian troops to suffer -40C in high attitude with poor equipment by nothing than their original deployment in Tibet and Xinjiang.