I wouldn't say christians aren't to read the bible at all.
2 Timothy 3:15,16 makes the point that the bible makes one wise for salvation and fully equips one. So a bible-believing christians would not want to omit to read the bible.
2 Peter 3:2 admonishes christians to remember what was previously spoken by the prophets as well as Jesus' commands given through the apostles. A christian would certainly want to refresh his memory on these matters by reading them in the bible periodically.
James 1:22-25 stresses the need to diligently apply what God's word says. That would certainly mean having to "peer into it" or study it. So there is no getting around the fact that christians ought to read - no, study - the bible.
But you would notice that James 1 talks about "hearing" the word of God and not "reading" it. I think the reason for this, and the reason for the absence of statements explicitly telling christians to "read" the bible, has to do with the fact that there were relatively few complete copies of the bible back then. Or, to put it another way, the average christian did not have a personal copy of the bible. That was a time before the invention of mass-printing and all copies of the bible had to be reproduced by hand - a time-consuming and expensive task. Thus only the religous leaders and some wealthy individuals had personal copies of the scriptures in their entirety. So the people depended on the religous leaders to read the scriptures to them at their weekly visit to the temple or synagogue. The situation was likely similar for early christians of the first century. They probably had a communal bible that was read from at congregation meetings. Few christians would have had personal copies. This is why Timothy was exhorted to apply himself to public reading at 1 Timothy 4:13; and this is why James 1:22 talks about hearing the word God. I'm saying all of this to say that the absence of explicit statement telling christians to read the bible is owing to the logistical reality of the times when personal copies of the bible were the exception. It's not owing to the reading of the bible being unnecessary (for they did have to "read" it with their ears when they attended meetings).
Having said all of that, I think that JWs take an almost superstitious view of the importance of reading the bible daily. Unless you're an ancient king or judge who has to be ever on guard as to what the law says, you probably don't need to read the bible daily. JWs see reading the bible as a spiritual end in and of itself and read the bible for the sake of reading it, as if the mere act of reading the bible makes them holy or righteous in God's eyes. There is the goal of reading the whole bible in a year by reading so many chapters a day. Is that really a worthy goal? Is it wise to read all the bible in a year just so you can claim to have read all the bible? So you see it becomes all about accomplishing a feat and reading the bible becomes an end in and of itself instead of the means to an end. Reading the bible becomes a legalistic religious work to make one feel righteous - all while they are distracted from what should be the ultimate reason and benefit from reading the bible. So JWs have a similar attitude to completing chapters in the bible as they have to counting hours in the ministry.
