All the Japanese IPTV streamers work that way and it's a very natural integration.
You can watch the current streaming, or just go back in the calendar to whatever you want in your selected channel. And the past items flow from one to the next as if they were live. You watch through last Monday's 8:00 p.m. one hour show and it falls through to the 9:00 p.m. show of that day as if it were last Monday.
There is no annoying "recording" workflow; the full content is just there going back about a month.
The organization concept that certain shows occur regularly at certain times is very useful; it won't go away that easily; it remains useful when you're able to navigate through the calendar.
Japanese TV, as such, is pretty great though. It is not replaceable in any way by other forms of content.
E.g. families with small kids depend on the NHK E channel; there is no equivalent content. Everyone knows what that is; it's not ignored, like BBC or PBS.