It is only the chief that is taxed, not every clan member.

Often the Konung will arrange  the timing of the tax, to get maximum benefit from it.

You also have to play smart.

10% of 0 is 0 

If the chief of your clan has no silver, there is no tax to pay.

How can you "play smart" when you can't be online right after cvc end ?!
Players are all over the world, there are some that can be online more easily than others right after receiving cvc gains.
That's just what Basile is saying.

I had assumed that the chief was logged in in order to know that the tax came 20 minutes after the reward was issued, but this might not be the case.

However,you have answered your own question.

The fact that players are distributed throughout various time zones is the solution, unless all the clan is in the same time zone,and its an inconvenient time for the chief to avoid paying the silver tax, in which case a solution is more challenging.

A system message saying the reward is in,is not going to help, if the levy has already been taken,

and avoiding that would require the reward to be paid at different times, which would be difficult to  accurately maintain where the chiefdom is subject to change.

It would be challenging for a dominant clan to organise the levy to fall at this particular time.  That is for a number of reasons but primarily because to do so the clan must defend the PoP for a period of time during the CvC event.  Defending the PoP against much weaker competitors is easy enough but defending it against a CvC opponent (who is likely to have equal power) is quite another matter.  In fact it is a losing game.

In practice clans do not do so.  The PoP is left undefended during CvC and so the first levy once it is taken back into the dominant clan's hands comes a day after CvC has ended.