It seems more likely to me that there is some sort of objection to 'web clients' because desktop clients are the 'one true way' to access IRC. Note that I say this as someone that prefers to IRC from console clients like BitchX and irssi through a screen session.
I known plenty of people that have used IRC that don't 'know' what IRC really is other than how to use the graphical elements of their client to access things for the 'chat room'.
And in this case you are not 'providing' IRC accessibility. Mibbit is providing that. The only thing that you're doing is restricting accessibility by banning Mibbit. Now if Mibbit were banned by default on your IRCd and you had to 'grant access' to them you could argue the other case.