It must be noted that the first sentence we find in the question (Il a dit qu'il « était le meilleur joueur du monde » hier soir.) and all sentences of that type do not have an equivalent form in standard French; if we find them nowadays it's on the count of their being borrowed recklessly from English, especially by headless journalists in need of attention; the guillemet is used to introduce in a sentence the very words someone has pronounced or written. The wish to keep to the flow of a usual sentence and the wish to show as faithful the reporting of the words heard or written cannot be reconciled entirely on the level of keeping the language simple. The form, morever, if it can be acceptable for straihtforward utterances, leaves a taste of adulterated grammatical techniques that is somewhat unpleasant. If it seems satisfactory in the way of having the best of both world when the language is kept plain, it's different matter when this is not so.
One will notice that a change of tense is necessary.
present in utterance → past in reported utterance; in French : "présent" in utterance → "imparfait" in reported utterance but not always, also "subjonctif présent"
Let's look now at a more complicated case in French;
Il a dit « faites attention, il y avait des balises mais il n'y en a plus! ».
In the English form we have the following as a first possibility;
Il a dit qu'ils « fassent attention, il y avait des balises mais il n'y a en a plus! ». (note that the "subjonctif" is needed)
It's not clear where the elements are situated in time; we can try this other option, in which we apply the time shift generally;
Il a dit qu'ils « fassent attention, il y avait eu des balises mais il n'y en avait plus! ».
In the utterance the beacons are reckoned to have been in place up to the locutor's time of speaking; that is not any more what the tense "plus que parfait" communicates; the new understanding is that at some point in the past there were beacons; one can get back at the true utterance, I would believe, but situating things at their proper time in the past is just not a matter of fact thing.
Another case; the speaker says "Faites attention, il y a eu des balises mais il n'y en a plus!". It seems that we must write this:
"Il a dit qu'ils « fassent attention, il y avait eu des balises mais il n'y en avait plus! »".
We cannot really tell what has been said.
All of this shows that with a small increase in complexity one is thrown into mind boggling considerations of tense correspondences and that it is much better to stick to the traditional form. The alternative to this added complication in the means of expression is to restrict its use to simple utterances and to revert to the traditionnal form for more complicated sentences; I, personnally, do not think that a desirable option: in doing so we introduce in the language one more weak technique, one more amputated means, which at that is nothing but a hybrid of two existing language techniques.
In conclusion I should say that this manner of using the guillemets (that is as in the question) is for now an unhappy English way of using quotes that shouldn't not be introduceded blindly into French, in which, fortunately, it's not been found up to the present day.