Alternative Heart Morphologies
Amphibians and some reptiles have a three-chambered heart, with 2 atria and a single ventricle. There are still separate circulatory pathways for the lungs and the rest of the body, but the oxygenated and oxygen-depleted blood mix in the ventricle, and are pushed at the same time to the lungs and body.
The disadvantage of this system is that it is less efficient at providing oxygen to the body than a four-chambered system, because the blood pumped to the body is a mixture of blood that has recently been to the lungs and blood that has not.
You seem to be suggesting another system, where there is just one serial pathway from heart to lungs to body, or from lungs to heart to body. This is more like the two-chambered heart of a fish: blood is pumped from the heart through the gills, and then to the body.
Lungs and body in series, and fluid dynamics
So why wouldn't this work for a mammal? It could, but gills and lungs are very different. The vasculature of the lungs requires a lot of pressure to get through, which is generated by the right ventricle.
Pressure is required to push fluids through a constriction (note that even a straight pipe is a constriction). After a fluid passes a constriction, there is always a pressure drop. That pressure drop is a function of the rate of flow and size of the constriction. If you give less pressure at the start, there is less pressure to "drop", and so you get a lower flow rate (note that this is nearly a perfect analogy with electricity, where flow is current, pressure is voltage, and the size of the tube is the conductance).
If you want to pass through a second constriction, you have to use pressure that is "left over" after the first drop: you can't use the pressure you started with because that was lost getting you through the first constriction. In reality, in a closed system, the total flow rate will be determined by the summed series resistance, so adding a second constriction slows the flow rate for the whole system, and you get a partial pressure drop after each.
If you put the lungs and body in series with each other, no matter which comes first, you need to have enough pressure left over when you pass one to get you through the next one, otherwise the flow will slow down. Blood in a human aorta has a mean pressure around 100mmHg, whereas back in the vena cava it is close to 10mmHg. That suggests you need about 90mmHg to get enough flow through the body's circulation. The lungs use a little less, but you still have a pressure drop of around 50-60mmHg. Therefore, you would have to increase the blood pressure by 50%-75% to get circulation through both lungs and body.
Higher pressures lead to more turbulent flow and take more effort to achieve. The four-chambered heart, with 2 ventricles, is simply more efficient.