IMHO there are several different ways to enter Starfleet.
The most common way is to enter Starfleet academy normally or enlist normally, with a minimum age of 18 or possibly 17 (Yeoman Tina "Lizard Girl" Lawton may have been only 17).
There may be a special program recruiting young geniuses to Starfleet Academy and that may be the program Wesley was tested for in "Coming of Age". I assume that all four of the tested were too young for admission the regular way, and too smart to fail the normal admissions test, so they should have been applying to a special and limited program that could afford to be very selective. Some characters might have entered Starfleet via that program - Kirk?, Spock?, one or both Chekovs?, Peter Preston (actor 18, age given as 14 in script and novelization)?, etc..
Similarly there may be different career tracks once in Starfleet and having to take a bridge officers test to stand watch might only be required for medical and psychology officers. Line officers might not always have to take the bridge officers test in order to stand watch, and/or might not always be promoted to commander after passing it.
note that Beverly Crusher was said to have been the "Head of Starfleet Medical" for a year.
WESLEY: That's all right. That will give me the time to finish some projects I have to do. It's going to be hard leaving the Enterprise.
PICARD: Mixed feelings for all of us. It's always difficult leaving any ship, just as it was for your mother
[Corridor]
PICARD: When she left to become head of Starfleet Medical. But going from one assignment to another is part of the life which you are choosing.
WESLEY: I know, but this isn't any ship.
PICARD: How true.
http://www.chakoteya.net/NextGen/127.htmUnless Picard was exaggerating Crusher's job, she was probably a temporary admiral while head of Starfleet Medical. And that seems a little unusual for 21st century navies but no doubt is rather ordinary in Starfleet.
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