the red tab that closes it off broke off
No need for a new tap or fitting, if you are not using it just close it by using a pair of pliers or grips to move the post, it only has to go through 90 degrees after all. Just make sure that when it feels like it is off, it actually is off, I have found on a number of occasions that these valves do not entirely shut the water off.
As the hose appears to be long enough I would just supply a bit more support for it, a couple of large hooks or even a couple of cable ties closed loosely (so you can thread the hose through) and secured to the underside of the work top should be fine. Don't forget that if you bring the waste pipe over anything sitting in front of it will stick out further than it might otherwise do.
I think (its a while since I looked at it) the hose from my Miele washer exits from the top of the machine
Al