Technical: SNI breaks older browsers (IE on XP, older mobiles). Also requires a modern (ish) OS on the server, if using CPanel, which like, every shared host in the world is pretty much.
User Education: Explaining exactly which clients will be broken is hard, etc.
#2 is basically gone now, because the users who would be broken are broken by the removal of SSLv3 and insecure crypto anyway - so we've now already killed them and can SNI without having to explain anything complicated to the user.
It was, but one that is becoming less and less relevant finally.
The problem comes from not knowing the host name until after the SSL/TLS channel has been negotiated, so the server doesn't know which certificate to hand out unless there is only one that is relevant for a given address.
SNI has been widely supported server-side and on up-to-date client applications for a long time, this is essentially a change to the protocol whereby the client tells the server which site it is looking for as part of the SSL/TLS negotiation so there is no problem providing the right certificate during that process. It has been supported by cPanel since the beginning of 2014, and presumable the competing shared host control panels too, so if your host is up-to-date then SNI should be available (if your host is running a 2014-or-before build of the control panel, move host now as they are likely vulnerable to a variety of security issues).
Unfortunately "up-to-date client applications" has been the sticking point: no version of Internet Explorer running on XP supports SNI, nor do older mobile browsers. Your site would still work on those older browsers but the user would get a security warning on each visit which is not desirable. Browsers that old are getting less common, and are getting locked out already because of protocol/cypher changes intended to mitigate recently discovered attacks against SSL/TLS, so SNI is an increasingly practical option (how practical is your decision: can you afford to potentially scare off that remaining small percentage of potential users? Personally I say "definitely yes, people that out-of-date are a nightmare to support anyway so my competitors can have them").
Of course IPv6 also fixes the issue too, but adoption of that lags behind the adoption of SNI compatible browsers by a long way.