That does not imply that there is enough testing. That just implies that your doctor or health care system is unwilling to test someone in your circumstances. "Not doing enough testing" is not referring to just test kids not being available, it is also referring to health system policies and willingness/unwilingness to test people.
We attempted to get tested and they said no, not worth the bother.
That seems contradictory to me.
If you're in an area where neither of those things apply then, for a child a test isn't going to change anything. It's not going to change their treatment. If it's positive they'll tell you to isolate, but if it's negative, there's a high enough false negative rate, and there are enough cases that it's still very likely they have it, so you'll still want to treat it like they do.
Not necessarily, it could mean that the people that need to get tested aren't all getting tested.
Japan did very little testing and has a relatively low death rate so far.
The reason is not that some people are not willing to get test. The reason is also capacity, aldo policies on who is allowed to get test etc. Your case is literally someone who should be tested and was not, hence "not enough testing".
Here, I can get comertial test if I am willing to pay. No one will send me home. And if I was in contact with covid infected person, I get test for free (it is mandatory). And if I live in household with sick person, I have mandatory quarantine.
Sounds like where you live is no contract tracing and thus not enough testing.
It can also be the case that insufficient testing is available in your area.
I know a group of people that were tested because of potential exposure (they were tested based on 1 person reporting symptoms; the additional tests were done prior to that result coming back).