TL;DR: There is nothing technically wrong with ECC; in fact it has better bandwidth and server CPU load than RSA (but higher CPU load on browsers, thanks @CodesInChaos), but for historical reasons it just sorta never got adopted.
Disclaimer: I work for Entrust Datacard
Entrust is one of the publicly-trusted CAs, and we have separate publicly-trusted PKIs (root + intermediates) for RSA SHA-1, RSA SHA-2, and ECC P-384. See the Entrust Root Certificates Download page.
The compatibility table on that page also answers your question about why ECC never really took off: clients / browsers / OSes were slow to adopt ECC. There was a time period where NIST and other agencies were urging people to move to ECC, but website admins were hesitant because of backwards compatibility issues. Now, even though virtually everything understands ECC today, it kinda just never took off.
Also, the NSA had been one of the big pushers of ECC, but in August 2015 they released a statement (source: wikipedia):
"Unfortunately, the growth of elliptic curve use has bumped up against the fact of continued progress in the research on quantum computing, necessitating a re-evaluation of our cryptographic strategy." NSA advised: "For those partners and vendors that have not yet made the transition to Suite B algorithms, we recommend not making a significant expenditure to do so at this point but instead to prepare for the upcoming quantum resistant algorithm transition."
So anyone who was still contemplating migrating from RSA to ECC now had zero reason to spend the effort.