English departments often incorporate a library lesson or silent reading lesson in their schemes of work. Why don't MFL departments?
Well, firstly there are severe time constraints on teachers to get through the syllabus in time frames that are already too narrow. It is not unusual for classes to have only two or three contacts per week (sometimes even less), so teachers feel reluctant to devote time to independent reading.
Secondly, there is a dearth (total absence?) of suitable reading material which would provide appropriate comprehensible input. There used to be. I and many colleagues made good use of the Bibliobus series in French. These were illustrated graded readers which students could work through on their own. I recall a perky and charming Year 7 girl asking me, as they all read silently, "Don't you get bored,sir, when we're reading?"
Thirdly, I doubt whether many teachers really buy into the claim that silent reading leads to significant gains in second language acquisition. they feel that they are only doing their job properly when they are providing input themselves and directing the learning. I bet you no teacher would plan a silent reading lesson for an Ofsted inspection either, even though it would be a perfectly good lesson. (That tells you something about Ofsted.)
If schools allocated enough regular time to languages, at least four contacts a week, and if publsihers produced the right material, either in book form or online, then I would still advocate sessions of sustained silent reading. In the end, whilst I don't necessarily go along with Prof Krashen is every respect, I still think acquisition comes down primarily to quality input.