Did Cato the Elder think destruction of Carthage was more important than anything else for Rome?

No.

We see an inference made in English language about an interpretation in English language over a single word and how to interpret it when the 'original' is supposed to have been in Latin.

That is: it is quite difficult to ascertain such a claim. At all. We only know the gist of it. We do not have the exactness of Cato's words.

Alas, the quote in question is debating one English translation from Latin that never was 'in Latin'.

Hu? Yes.

A few ancient writers re-tell the story of Cato the Elder saying something to that effect, constantly.

But we only have one ancient writer actually presenting something like a quote to us.

Plutarch.

In his Parallel Lives we find the only ancient quote to illuminate.

Problem?

Indeed.

It isn't Latin, but Greek!, and missing the contentious word altogether:

Πρὸς τούτοις φασὶ τὸν Κάτωνα καὶ σῦκα τῶν Λιβυκῶν ἐπίτηδες ἐκβαλεῖν ἐν τῇ βουλῇ, τὴν τήβεννον ἀναβαλόμενον, εἶτα θαυμασάντων τὸ μέγεθος καὶ τὸ κάλλος, εἰπεῖν ὡς ἡ ταῦτα φέρουσα χώρα τριῶν ἡμερῶν πλοῦν ἀπέχει τῆς Ῥώμης. ἐκεῖνο δ' ἤδη καὶ βιαιότερον, τὸ περὶ παντὸς οὗ δήποτε πράγματος γνώμην ἀποφαινόμενον προσεπιφωνεῖν οὕτως· "δοκεῖ δέ μοι καὶ Καρχηδόνα μὴ εἶναι". τοὐναντίον δὲ Πόπλιος Σκιπίων ὁ Νασικᾶς ἐπικαλούμενος ἀεὶ διετέλει λέγων καὶ ἀποφαινόμε νος· "δοκεῖ μοι Καρχηδόνα εἶναι". πολλὰ γὰρ ὡς ἔοικεν ἤδη τὸν δῆμον ὁρῶν ὕβρει πλημμελοῦντα, καὶ δι' εὐτυχίαν καὶ φρόνημα τῇ βουλῇ δυσκάθεκτον ὄντα, καὶ τὴν πόλιν ὅλην ὑπὸ δυνάμεως ὅπῃ ῥέψειε ταῖς ὁρμαῖς βίᾳ συνεφελκόμενον, ἐβούλετο τοῦτον γοῦν τὸν φόβον ὥσπερ χαλινὸν ἐπικεῖσθαι σωφρονιστῆρα τῇ θρασύτητι τῶν πολλῶν, ἔλαττον μὲν ἡγούμενος ἰσχύειν Καρχηδονίους τοῦ περιγενέσθαι Ῥωμαίων, μεῖζον δὲ τοῦ καταφρονεῖσθαι. τῷ δὲ Κάτωνι τοῦτ' αὐτὸ δεινὸν ἐφαίνετο, βακχεύοντι τῷ δήμῳ καὶ σφαλλομένῳ τὰ πολλὰ δι' ἐξουσίαν πόλιν ἀεὶ μεγάλην, νῦν δὲ καὶ νήφουσαν ὑπὸ συμφορῶν καὶ κεκολασμένην, ἐπικρέμασθαι, καὶ μὴ παντάπασι τοὺς ἔξωθεν ἀνελεῖν τῆς ἡγεμονίας φόβους, ἀναφορὰς αὑτοῖς πρὸς τὰς οἴκοθεν ἁμαρτίας ἀπολιπόντας. οὕτω μὲν ἐξεργάσασθαι λέγεται τὸν τρίτον καὶ τελευταῖον ὁ Κάτων ἐπὶ Καρχηδονίους πόλεμον, ἀρξαμένων δὲ πολεμεῖν ἐτελεύτησεν, ἀποθεσπίσας περὶ τοῦ μέλλοντος ἐπιθήσειν τῷ πολέμῳ τέλος ἀνδρός, ὃς ἦν μὲν τότε νεανίας, χιλίαρχος δὲ στρατευόμενος ἐπεδείκνυτο καὶ γνώμης ἔργα καὶ τόλμης πρὸς τοὺς ἀγῶνας. ἀπαγγελλομένων δὲ τούτων εἰς Ῥώμην, πυνθανόμενον τὸν Κάτωνά φασιν εἰπεῖν· οἶος πέπνυται, τοὶ δὲ σκιαὶ ἀίσσουσι.

Which is translated:

In addition to this, it is said that Cato contrived to drop a Libyan fig in the Senate, as he shook out the folds of his toga, and then, as the senators admired its size and beauty, said that the country where it grew was only three days' sail from Rome. And in one thing he was even more savage, namely, in adding to his vote on any question whatsoever these words: "In my opinion, Carthage must be destroyed."​h Publius Scipio Nasica, on the contrary, when called upon for his vote, always ended his speech with this declaration: "In my opinion, Carthage must be spared." 2 He saw, probably, that the Roman people, in its wantonness, was already guilty of many excesses, in the pride of its prosperity, spurned the control of the Senate, and forcibly dragged the whole state with it, whithersoever its mad desires inclined it. He wished, therefore, that the fear of Carthage should abide, to curb the boldness of the multitude like a bridle, believing her not strong enough to conquer Rome, nor yet weak enough to be despised. 3 But this was precisely what Cato dreaded, when the Roman people was inebriated and staggering with its power, to have a city which had always been great, and was now but sobered and chastened by its calamities, for ever threatening them. Such external threats to their sovereignty ought to be done away with altogether, he thought, that they might be free to devise a cure for their domestic failings.

4 In this way Cato is said to have brought to pass the third and last war against Carthage,​24 but it had p385 no sooner begun than he died,​25 having first prophesied of the man who was destined to end it. This man was then young, but as tribune in the army, he was giving proofs of judgment and daring in his engagements with the enemy. Tidings of this came to Rome, and Cato is said to have cried on hearing them:—

This leaves us with the reminder: since the exact words are not known, and the often found ceterum missing from the original Greek, most speculation about the exact meaning of this later interpolation seems quite moot.

What we can infer for sure is that Cato knew the old trick of 'repeating what's important' also known as redundancy.

A nice reconstruction of this invented tradition for 'a quote' is:

But even if Polybius in his complete text of Book xxxvi did quote the famous saying about the destruction of Carthage, he probably made no more of it in the dramatic and declamatory sense than did Cicero or perhaps even Livy. It was left for the changed conditions of life and literature of the Silver Age to give it this turn. We miss it, to be sure, among the remnants of the elder Seneca (about B.c. 55 - about A.D. 40) who made collections of these school exercises (Suasoriae now in one book, Controversiae now in five of the ten books). But even if he did include it, it was finally lost in the many condensations of his work which must have been once very much more bulky and contained items far more numerous than we have at present.

In the Silver Age after Seneca the Elder the elocutionary value of the phrasing seemingly became appreciated and insisted upon. Pliny the Elder and Plutarch had begun to form the tradition, while later Florus and Aurelius Victor give the finishing touches that made the tradition lastingly declamatory and almost sensational. It was thus that the phrase became familia modern English readers as a standing description of ender" and the ready formula of celebrated chauvinis goes even to the present day. In fact and feeling it was descriptive of Cato; its form was shaped by the life a taste of imperial times.

— Charles E. Little: "The Authenticity and Form of Cato's Saying 'Carthago Delenda Est'", The Classical Journal , Mar., 1934, Vol. 29, No. 6 (Mar., 1934), pp. 429-435

Also, you have been lied to about pretty much everything 'Covid' ;)