Except for it's never done. No Hindu believes Allah or Jesus is an Avatar of Vishnu.
It can't be. Vishnu literally means the "All-Pervading One". Allah is explicitly not Omnipresent. Jesus is Omnipresent but does not have infinite forms. So neither fit the bill. Even Buddha, who we Hindus associate as one of the Avatars of Vishnu is not the same Buddha as Gautama Buddha who founded Buddhism.
There's a rich history of panentheism in both. Ibn Arabi, probably the most famous Sufi scholar was known for defending it ("[God] sleeps in the rock, dreams in the plant, stirs in the animal, and awakens in man"), Meister Eckhart and the Rhineland mystics held similar views etc. Of course that often got them into hot waters with Orthodoxy but it certainly existed and continues to exist.
A lot of these traditions are much more flexible than whatever is the dogma or institutional version of the time suggests. David Bentley Hart's writing on Vedantic traditions and Christianity is worth reading
This is literally how Islam got to places in Asia and spread. It was through Sufis who acted as initial points of contact.
> A lot of these traditions are much more flexible than whatever is the dogma or institutional version of the time suggests.
This is only so when it comes to missionaries in each religion. Be it Evangelists saying Vishnu is Avatar of Christ (we literally have Christ idols shaped like Vishnu with multiple arms and sitting on a lotus in a Church in India), Sufis singing Krishna is same as Allah or heck even ISKCON claiming Christ is an Avatar of Krishna /Vishnu. That does not make "traditions flexible". It's just a cheap tactic to increase followers/harvest souls. These converts are typically fluid in their own convictions and can easily be swayed one way or the other based on reasons unconnected to the Religion/Dogma/Philosophy itself. Reasons could range from political, monetary to finding fame/popularity within a new sect/group/cult/ideology.
Very few are seekers of Truth.
That's not mutually exclusive. You're certainly right that religions spread this way, but it's also what they become. Christians today are Pauline Christians more than anything else. The abolition of circumcision and dietary restrictions, household code, 'render unto Caesear' etc were obviously tactics to convert Gentiles, but once you had all the gentiles, you had genuinely altered what the faith is. They never went back to the radicalism of Jesus.
Buddhism is also an interesting case. By the start of the 19th century there wasn't strong Buddhist practice in India. When it was rediscovered it was heavily shaped by China, Tibet and Japan and had done a round about turn.
Same with modern Evangelicalism and Catholicism. Very successfully spread through indigenous cultures in LatAm and Africa but now the influence goes the other way, with real impact on the churches, just look at the last two popes. I don't think this deserves to be categorized as cheap tactic, that's very much what religion is.
I'm sure you are right that many/most Hindus would consider this incompatible with their belief system (and even more sure your average Christian would think he was missing the point of accepting Jesus as his Lord and Saviour), but this sort of syncretism definitely has been attempted and found its converts, probably rather more of them than Akbar's attempts to combine Islam and Hinduism.
I'm also reminded of this mostly unrelated comedy sketch :) https://www.youtube.com/watch?v=8tw7LIykvBw
For central authorities of the Catholic Church, and Hindu theologians, this is all heretical, but the Catholic Church allowed for flexibility around traditions as it expanded. One big example is that many of what are considered Christmas traditions in northern Europe (the Christmas tree being an especially prominent one) are hangovers from traditions that predate Christianity.
This man has definitely been to Kerala.
That's exactly the point of why he even attempted to have this conversation with you. It was to convert you to Hinduism.
I explain it here: https://news.ycombinator.com/item?id=49620794
> some interpretation of the Vedic texts, which doesn't actually entail all that much
If you studied the Vedas you would know why this issue exists in the first place. The Vedas talk of One Supreme Brahman who should be given Havis during a Yajna. The dispute between the sects is on who the Supreme Brahman is identified as. Every name in the Vedas is but a name of the Supreme. Vyasa composed the Brahma Sutras to unravel it. But then, as is the case with Dharmics, we have around 30+ commentaries on the Brahma Sutras itself. This, coupled with commentaries on Prasthanatrayi as well as interpretation of Puranas created sects.
All of them come within the umbrella term of Hinduism as all of it is Dharmic and native to the Indian subcontinent.
What do you mean by "local religion"?
We also have Christian nationalists in the US, for a more politically active flavor.