I'm not an expert on this one, but my gut feel is that linking rivers will probably greatly change and likely destroy ecosystems. This idea has been floating round in India from the time I was in my teens or a bit later, and there has been both support and opposition to it from various groups. IMO, at least the wildlife - both flora and fauna, not to mention micro-organisms, will drastically change (and some of them will likely be heavily affected), due to the sudden physical and chemical changes (like differing levels of dissolved salts and other chemicals, silt, salinity or freshness), of some rivers versus others, when they are linked together.
A better idea is probably much more intensive and extensive rain water harvesting, building thousands more ponds and "tanks" (term here for larger ponds) for water collection, water usage reduction (where it makes sense), water conservation, and majorly, afforestation.
For afforestation (and preventing or reducing deforestation, first), it is well known that forests increase rainfall and also moderate and buffer its dispersal once it is on/in the earth, and reduce/slow down its discharge from the ecosystem, thereby proving more water throughout the year, even in dry seasons.
See the book The Forest and The Sea by Marston Bates [1].
I had read it as a teenager. A very good exposition of the many interacting factors in natural systems like forests and seas, symbiosis, and all the benefits they provide for humans.
[1]:
https://en.wikipedia.org/wiki/Marston_Bates
https://www.britannica.com/biography/Marston-Bates
https://www.goodreads.com/author/show/177375.Marston_Bates
https://www.goodreads.com/book/show/1481687.The_Forest_and_t...
https://textbookhistory.com/marston-bates%E2%80%99-moment-of...
From above link:
[ "Though Rachel Carson is usually credited for raising the public’s awareness of ecology, as Marion Clawson noted, it was Marston Bates’ 1960 book, The Forest and the Sea, not Silent Spring, that made “ecology a household word.” ]
https://www.amazon.com/Forest-Sea-Marston-Bates/dp/B000P0D61...
The Indian government, NGOs and corporates could look to countries like Israel (which seem to have done wonders in the desert with drip irrigation and many other techniques - I've only heard a bit about it, some from my grandfather, who was a scientist in the field, and visited there), to solve the water issues and also improve resource usage / conservation and also better and more agriculture production, all at the same time.