Electric current is a flow of charge. The electrons, ions, protons and positrons that carry charge can move in their own ways, appropriately and in either direction at the same time even. The charged particles may have their own physical interactions and effects, but it is the flow of charge that causes electrical actions. The charge carriers are particles that have charge. They do not rush around the circuit at close to the speed of light.
The direction of current (charge flow) matters in that it determines the polarity of magnetic field produced, one of the inherent electrical effects. It sometimes helps to have a sign for current flow direction so mathematical solutions end up with the right polarity of the voltage or magnetic field. Therefore a convention is used. It doesn't really matter which direction current flows. Ohms law has no sign because resistors are not polarity conscious. The drawing with little minus signs is therefore misleading. We are not talking about water flowing in a pipe, and we are not dealing with forces caused by static charges. The charge moves at close to the speed of light, while the electrons (and other charge carriers) move as slowly as meters per hour.
The polarity of the voltage does matter, because many components are polarity conscious. My advice is ignore direction of current flow, at least most of the time. Focus on the polarity of the voltage. Current direction mostly does not matter, and even the polarity of a magnetic field can be determined by voltage polarity.
As far as current regulation is concerned in the laser diode circuit, the diode and the current regulator are in series across the battery. The other bits are just incidental, to set up the proper behavior. As the current is the same in all parts of a series circuit, it doesn't matter what order in the circuit the regulator is, relative to the diode. What does matter is the polarity of the laser diode and the regulator. Both would be destroyed in a few microseconds if they were powered the wrong way round.
It doesn't help to think of the charge propagating along the wire too much, although this does happen. It seems more likely to me that it propagates from both sides of the supply, meeting in the middle. During this time the build up of current is controlled by the inherent stray capacitance and inductance and resistance, and various parts of the circuit itself. It is a transient state, taken over by the settled state of the whole circuit eventually. In this case the capacitor is important because it slows the rate that voltage can change across the laser diode. It dominates the transient state, so any regulator transients (overshoot) as it starts up cannot damage the laser diode. By the time the capacitor is charged to the laser diode voltage the regulator is in control. This is because the laser diode is unusually sensitive to brief over voltage/over current. The charge can propagate around this circuit in less than a nanosecond. The capacitor charges in milliseconds. The regulator works in microseconds.
Regulator and laser diode position in the circuit...
Circuits are usually drawn with a positive or negative ground to clarify how external interfaces (connections to other devices) are made. Your example has no external connections other than the supply, so it doesn't really need the ground to be defined. There are issues though. The metal case of the laser diode is connected to the negative supply, so if the positive supply is grounded, its a short. Usually the case is attached to a metal heat sink, so issues of heat sink insulation are avoided by making the negative supply ground, and drawing the components in the order they are. Yes it could be rearranged to swap the order of the laser diode and the current regulator but that would be "unconventional" and require the laser diode case to be insulated. Shorting its case to the negative supply (grounded in a car) would cause its instant destruction. As it stands, shorting any part of the diode has no destructive effect. The regulator is on the active side, so its heat sink has to be insulated, but it is self protecting.
Probably early educators need to clean up their act. While they need to simplify for early classes, they do not need to make it so it needs to be unlearned later.