Why we need to use a modulus in public cryptography?
The DH, ECDH and RSA problems require modular operations. They are fundamental in the problem space that they operate in (where problem is a mathematical term). Please go through an introduction of number theory for cryptography to understand why - or take it for granted if you don't want to delve that deep.

Besides that, performing exponentiation on large numbers is definitely more troublesome without modulus than with a modulus. Imagine raising a number of, say 2048 bits to the power of a private exponent of 2048 bits. The resulting number would have a value of about $2^{{2048}^{2^{2048}}}$. Now I don't know about you, but I'm pretty sure that number is too big to handle. Modular arithmetic is much more efficient than non-modular arithmetic as well; you can easily get a feel for this yourself, what's the last digit of $2459 \times 9823$ or $9823^4$?

Finally, we don't need to have an asymmetric primitive such as RSA to handle large plaintext/ciphertext; we can use symmetric primitives for that, creating a hybrid cryptosystem. For instance we can use a cryptographic hash (such as SHA-256 or SHA-512) to create signatures together with RSA-PSS. And we can use a symmetric cipher together with RSA-OAEP or RSA-KEM to create a secure cryptosystem. That way we can handle larger plaintexts / ciphertexts without having to deal with the size limitations of asymmetric primitives.