There's nothing cryptographically broken about RSA as a cryptosystem per se, but implementing it correctly is difficult. There have been multiple revisions to the standards for RSA over the years in response to various attacks. The current standard is PKCS#1 v2.2 (https://tools.ietf.org/html/rfc8017) and we should use RSAES-OAEP / RSASSA-PSS as primitives.
However a PKCS#1 1.5 compatibility mode with fixes for Bleichenbacher's oracle is also present in this standard and also specified in TLS 1.2 for compatibility reasons. In order not to provide an oracle, padding and other properties must be verified and a random premaster secret returned on failure instead of an error message, see: https://tools.ietf.org/html/rfc5246#section-7.4.7.1 . Note that under the other techniques, there are a lot of caveats and remarks.
Keys must also be carefully generated, as demonstrated by ROCA. In this case a specific format of primes was used to make prime number generation faster, which unfortunately also happens to be vulnerable to attack by Coppersmith's method. Any such key is weak in the sense that the private key can be recovered.
This is a quick overview. There's a lot of literature on attacks on RSA, particular parameter choice etc. So one argument against its use is the many issues that exist and the fact that even for experts, implementing it correctly is not easy.
The other point is that, compared to Elliptic Curve cryptosystems, RSA is (usually) more expensive as an operation and key generation is certainly more expensive. As a result, elliptic curves work better in constrained environments like on smartcards. If you want to do forward secrecy by periodically generating and throwing away random keys, elliptic curves can do this more efficiently - RSA by contrast would be slow.
However a PKCS#1 1.5 compatibility mode with fixes for Bleichenbacher's oracle is also present in this standard and also specified in TLS 1.2 for compatibility reasons. In order not to provide an oracle, padding and other properties must be verified and a random premaster secret returned on failure instead of an error message, see: https://tools.ietf.org/html/rfc5246#section-7.4.7.1 . Note that under the other techniques, there are a lot of caveats and remarks.
Keys must also be carefully generated, as demonstrated by ROCA. In this case a specific format of primes was used to make prime number generation faster, which unfortunately also happens to be vulnerable to attack by Coppersmith's method. Any such key is weak in the sense that the private key can be recovered.
This is a quick overview. There's a lot of literature on attacks on RSA, particular parameter choice etc. So one argument against its use is the many issues that exist and the fact that even for experts, implementing it correctly is not easy.
The other point is that, compared to Elliptic Curve cryptosystems, RSA is (usually) more expensive as an operation and key generation is certainly more expensive. As a result, elliptic curves work better in constrained environments like on smartcards. If you want to do forward secrecy by periodically generating and throwing away random keys, elliptic curves can do this more efficiently - RSA by contrast would be slow.