Why let a pressure canner vent, before putting the weight on?

In canning, you have to be very sure that you have produced the exact conditions needed for the process to work properly. If you deviate a bit, it can happen that bacterial spores survive. These "exact conditions" mean not only "keeping temperature X for time Y". You have to make sure that proper heat exchange happens where it needs to happen.

In the case of the venting step, it is needed to ensure that your canner is voided of air entirely. When the water starts to boil, it turns to a gas and this gas mixes with air. If you do it in a sealed container with a small exit (like a canner), the steam can push out the air, but it needs some time to do so. This is why recipes are designed with a vent time at the beginning. Air pockets would interfere with the heating process. Only after you have achieved the canner-full-of-steam-without-air condition, can you start building the pressure.

You can see a quote for it here:

Then allow the steam to escape for an additional 10 minutes to vent the canner. This step removes air from inside the canner so the temperature is the same throughout the canner

Which brings us to your other question, how to know when to stop. The simple answer is: do exactly as your recipe says, and don't deviate. If it says "10 minutes", put a timer and let it go for 10 minutes. OK, you do have some wiggle room in the "longer" duration, if it takes you 3 minutes to manage to put the weight on, your food is still safe despite it having been vented for 13 minutes. But don't go below what the recipe tells you. And also note how sensitive the recipes are to change - canning the same recipe in a different size of jar already changes the safe times. So, really, don't try to second guess your recipe, do as you are told.

Have a look at this: Air and Steam Mixture

Mere pressure at up to 30 PSI ('15 PSI' on the gauge is really ~30 PSI absolute at sea level) will do nothing to pasteurize your food. It's high heat that does so.

Following the venting procedures on a Presto left approximately a 20% air mixture in with the steam, according to my own measurements and calculations. At that amount of air, the '10 PSI' on the gauge in reality equates to about 109 degrees Celsius (228.2 Fahrenheit) and not the theoretical 113C / 235F that is often claimed for '10 PSI'; at '15 PSI' it's 115C / 239F; not the theoretical 121C / 250F.

And that's after letting it vent for 10 minutes.

If you don't vent it, you're increasing the air portion in the canner even further and thereby lowering the temperature from what has been shown to successfully pasteurize the food.

The recipes have been tested using actual canners, so I am not too concerned that my temperatures are lower than the theoretical ones.

But you may barely get above water canning temperatures if you don't vent: air mixture drastically lowers the temperature at a given pressure.