I pulled and rebuilt a Series 612 apron that is probably of same design. Here are some bits of info that may help you in thinking through this.
First, pulling the apron/saddle is not hard but it does require a hoist or such for support. To remove the apron, you'll have to remove the control rods and leadscrew as you already know. You 'can' slide the apron/saddle off the end of the bed without removing the rods but I don't recommend it. You’ll need an engine hoist or forklift to go this route plus an additional hand to keep the system from binding up when you slide it off. (I did it this way the first and it's much easier to remove the rods first.) The leadscrew has a bearing assembly that is held on by a spanner nut. You'll have to slide the leadscrew out of the gearbox to get to the spanner nut. As you said, slide a piece of round stock through the highest hole in the apron, apply tension, then remove the 8 or so cap screws holding the apron to the saddle. Lower and pull away the apron (you have to clear the carriage wheel spur gear from the rack gear).
Second, the apron/saddle oiling system....hmmm.....not the easiest to describe nor troubleshoot. On my 612, the oiling system is run off the rapid traverse while on my 62, it is run off a cam on the carriage wheel (no rapid traverse). What system do you have? This oiling system is not a simple group of copper lines but instead is a dynamic system that oils certain parts when certain levers are used. For instance, there is an oil line that oils the half-nut, but only when the half-nut is engaged. Only when the oil ports in the half-nut are lined up via engagement will the oil flow. In addition, the crosslide feed and carriage feed gears are oiled via 'valves' that are operated via those handles. From memory, I remember the following items oiled via this system: the cross slide and carriage feed gearing/bearings in the apron, the half-nut, the worm gear, the shoe that slides on the bed face, the cross slide ways, the cross slide handle, the front vee and back flat ways (bearing on the bed, and the inside flat way that doesn't bear on the bed. The coiled tubing you show above is nothing but a simple pressure regulator, commonly called a capillary tube.* Underneath the saddle there are bijur flowlimiters that should be replaced (don't bother trying to clean them). Again, this is not a simple system but you should be getting oil out of each port on the bed ways at all times. If not, check the flowlimiters, if you can get to them.
Oh, by the way, the Monarch manuals don't describe nor show the oiling system lines. You'll have to decipher the oiling lines yourself.
* A capillary system is just a long, thin tube that provides a lot of resistance to a fluid pumped through it. The length and diameter of the tube coupled with the viscosity of the fluid provide a certain backpressure, thus regulating system's fluid pressure. You should have some oil dripping out of your capillary tube during normal operation.