I've been struggling to find a good analogy to explain why this simplistic reasoning is wrong. Let's see...
Imagine that an alien has landed on Earth and struck up a conversation with you. He/she/it comes from a world where vehicles may or may not have wheels. Let's say that wheels are the exception there rather than the rule. This alien might then look at our world and go, "Hey, a car with wheels!" "Oh look, another car with wheels!"
After a while, you might feel like pointing out gently, "Look, you don't need to say 'with wheels', OK? All cars have wheels."
And the alien might go, "I do not understand your logic, human. This is a car. It has wheels. Therefore, car-with-wheels!"
And you go, "No, no, no. All cars have wheels. That just makes them cars. In fact, if someone tries to sell you a car without wheels, they're ripping you off."
(The alien looks extremely puzzled and decides to travel to a less confusing world.)
That's the best analogy I can think of. All Indian movies have songs in them. That does not make them "musicals". They're just movies. Still confused? Look, it's a different genre, OK? Just leave your preconceptions at the door before entering, because you're the alien here.
Indian movies are often referred to as masala movies because the best ones are a blend of different "spices" - some drama, some humour, some action, some romance, some songs and dances, some family values, a moral, etc. There's something in it for everyone. That's how you enjoy Indian movies - savour the entire mix of spices that they offer. Songs are absolutely de rigeur. The masala is incomplete without them.
So remember, if you've just seen an Indian movie without songs in it, ask for your money back.