No not really. It's a thing you can do just because you can, like writing a book without the letter "e". Closer to the context of computation, it's like building a computer using only NAND gates. Or, given how restrictive the S combinator supposedly is, using only AND and OR gates. It won't make an efficient computer because your design is convoluted to all hell to make up for your choice to never use a NOT gate. Even the one made from NAND isn't efficient, because even though NAND can make any circuit without being too convoluted, it's not the most efficient way to make all circuits.

If anything, this would be a less compact representation.