Setences like "If the jungler can do without blue then so can the mid laner" betray a fundamental lack of understanding of what blue buff does for different champions on the map. Blue buff on a jungler is worlds apart from blue buff on a laner and the jungler being able to be effective without it doesn't mean the mid laner can be, particularly as your mid laner in this scenario is very possibly trying to match up against an enemy mid who DOES have blue access, and that's a cripping handicap for the one devoid of the buff.
Blue is a situational power buff in ganks (slightly more AP, lower cooldowns if multiple rotations are necessary) with a bonus of situationally increased map uptime (going back for mana isn't a thing with it). It won't always make a difference, because it can. For mid laners it means hugely empowered ability to harass and shove for a long period (= inflated chance of kills and roams, ability to ensure enemies such as Zed cannot outpush and auto-win the lane). Blue buff fundamentally changes the gameplay of a mana-using mid laner in a hugely empowering way, blue buff makes hardly any difference to a jungler beyond the first one. On top of that, "one jungler has blue and the other doesn't" makes very little difference to the jungle matchup beyond first clear whereas in mid lane uneven blue buff distribution can turn the lane.
You need to put a lot more thought into stuff like buffs because incredibly basic arguments such as "It's not that big a deal for the jungler, therefore it can't be a huge deal for anyone else" are just flat out wrong. You should have asked yourself a long time ago why so many mid laners felt like the blue was so essential to them and found at least one of the varying reasons behind that phenomenon. Of course, there are situations where the jungler should keep it, but that basically boils down to player skill and the question "Will they be able to avoid giving it away?". If your mid laner seems competent and capable of staying alive for the duration of the buff then you should absolutely give it to them.