As others have said, the rule you cite is for deciding which country should evaluate a traveler's Schengen visa application. For travelers who do not require a Schengen visa, the rule is irrelevant.
As you note,
embassies are very picky with this
But not "at least when processing visa applications"; they're picky about the rule only when processing applications, since that's what the rule is about.
The remainder of your question depends on incorrect assumptions or inferences about the Schengen visa rules.
it is well-known that you normally need to arrive in the country where your Schengen visa was issued
That's not true. It is a common misconception that you need to arrive in the country that issued your visa. In fact, such a rule would be completely illogical in the face of the visa-application rules, and inconsistent with them.
Consider the example of someone who plans to fly to Rome, spend a couple of days in Italy, then travel to France and spend three weeks there, then return to Rome for a couple of days before flying back home. Under the visa rules, that person must apply at the French consulate. All other consulates must refuse the application.
A rule requiring the person to enter through France would be completely incompatible with the rule requiring application to the main destination country. If the Schengen area wanted to have such a rule, then the rule for determining the country of application would be "you must apply to the country where you will first enter the Schengen area." But that's not the rule, of course.
otherwise you will be changing your itinerary
See the previous example. You would be changing your itinerary only if you submitted an itinerary where you enter your main destination directly. But these are not the only acceptable itineraries. You can apply with an itinerary where you enter through another country, and then when you arrive at that country, you are not in fact changing your itinerary.
For example, when having a Schengen visa issued by France and then entering the Schengen area through Italy (since France it's not your main destination anymore and you applied through it before, it could be interpreted as if you wanted to go Italy from scratch instead of France, which is a visa fraud)
Again with the previous example: The Italian border guards can ask you to prove that France is your main destination when you enter, but they can't deny entry simply because you arrive in Italy with a visa issued by France. When you show them that your application presented an itinerary beginning in Italy, along with evidence of your onward travel to and sojourn in France, they will let you in.
So, if I decide to visit France for 10 days and The Netherlands for 5 days, do I need to follow the "rule" where I SHOULD arrive in the main destination country? (in this case, France) or I am free to go through The Netherlands first even if I am spending less days there (remember that border officers could ask for itinerary/hotel reservations).
Since there is no such rule, you don't need to follow it. As a visa-free national, you can follow this itinerary just the same as you would be able to follow it holding a visa issued by France. Remember, if you were pursuing this itinerary as a visa national, you would be required to apply to the French consulate even though you are first spending five days in the Netherlands.