Why are through holes PLATED in PCBs?
In order for your scheme to connect the top and bottom layer, TWO conditions must BOTH be met:
- The pad on the TOP must be accessible and must be soldered (separately).
- The pad on the BOTTOM must be accessible and must be soldered (separately).
In very many cases the top pad of a thru-hole component is NOT accessible because the body of the component covers it. So that is not practical.
In MOST cases there IS NO component lead at all where you need to via from one side to the other. Inserting short bits of wire and soldering BOTH SIDES is simply not practical even for manual assembly not to mention automated assembly as virtual all modern gear comes from.
It doubles the effort to require soldering to BOTH sides of even a thru=hole component lead. That takes double the assembly time, and greatly increases the chances of assembly error. It is simply not reasonable at any level.
Your first image is not completely accurate. The solder should also be sticking to the plating in the hole, and not just connecting to the top and bottom layers of metal. That is to say, plating offers improved mechanical stability and increased joint strength due to the much larger surface area available for soldering.
If you have plated-through holes, tracks on both sides of the board (and any internal layers) are connected without any further action.
Holes exclusively for this purpose are called "vias" and may be smaller than regular holes for component leads.
This makes manufacturing any board too complex to be single-sided, easier and usually much cheaper than otherwise possible, since no extra effort such as inserting jumper wires or soldering on both sides is required.
It also makes design and layout of double-sided boards much easier since you no longer have to strain to minimise the number of tracks on the "other" layer, or minimise the number of jumpers, or ensure crossings between layers aren't underneath components.
And that allows you to increase the board density and use smaller boards, cheaper enclosures, etc...
It also allows the PCB manufacturer to perform "bare board testing" of every one of these interconnections before any components are added - thus eliminating many defects. (Some PCB makers perform bare-board testing free of charge).
Plated holes give you all this before even considering how you actually solder a component to the PCB - though it offers advantages there too...