Will non-ECC algorithms like RSA eventually become too inefficient?
RSA will never become "impractical" until it gets broken. (in polynomial-time)

As the best current algorithm to break RSA is sub-exponential (general number field sieve / GNFS). According to Moore's law computing power grows exponentially however. So simply increasing key size will always be an option.

Beyond 256-bit security EC crypto just is significantly faster than "classic" crypto. For example, for 512bits, EC crypto is 64 times faster than classic crypto (compared to private key operations using RSA).

If you are interested in key sizes you should read this paper.