Nothing would better express the Traditional, Biblical, Patristic, notion of the Petrine Office, as happily defined at Vatican I, than the following reform in what is done at the inauguration of a pontificate.
THE NEXT CONCLAVE
The prelate who emerges elected from the next Conclave should instantly sweep away all the unnecessary and obsessive ritual flummeries dreamed up, I think, by Pietro Marini and first used at the Inauguration of Benedict XVI.
They should be replaced by the taking of a solemn Oath of Fidelity done in public. Analogies and formulae existed for this in previous ages, which might be used to supply textual materials.
For people who like ritual stuff, the Oath could be taken on the oldest Bible in the Vatican Libraries.
For people who like even more ritual stuff, or are fixated on the use of vernaculars, each paragraph, before the pope recites it in Latin, could be chanted by, say, a Jewish Cantor in Hebrew and a Byzantine Subdeacon in Greek and read by laypeople in any number of vernaculars.
For strange people who want even more ritual than that, the document could then be solemnly attested by Cardinal Notaries and sealed with lead.
For those whose affection for ritual amounts to a lunatic obsession, the document could, finally, be solemnly processed through the congregation, held aloft by the Cardinal Protodeacon in the popemobile, while the crowd hysterically shouted VIVAT IUSIURANDUM! Meanwhile, the pontiff would remain kneeling in quiet and humble prayer before the Altar. The popemobile could then be taken away, either for immediate ignominious destruction, or for sale at Sotheby's in New Bond Street, the money (including, of course, the auctioneer's commission) being given to the Poor and the Ordinariates.
WHAT SHOULD HE SWEAR?
Essentially, the new Pontiff should swear, in words drawn from Pastor aeternus of Vatican I, to hand down uncorrupted the Tradition which is from the Lord through His Apostles, the Deposit of Faith. He should swear to resist and to put down all novitates, tam in Fide quam in moribus.
He should acknowledge that, while he will indeed be the Supreme Legislator with full power to change the (human) Law of the Church, he will himself obey the Law and refrain from interfering with legal processes, particularly those relating to the trial, conviction, and punishment of clerical sexual predators who enjoy powerful curial protectors.
And a rather useful practical undertaking might be: "As We uncover evil practices and corrupt deeds and false teachings among Our Cardinals and their clientelae, We shall not allow Ourself to be deterred from dealing with them strictly and according to Justice, by any consideration of who supported or who resisted Our Own election."
To preserve the poor silly Media from their inveterate temptation to assume that a pope has or ought to have a "programme", the Oath should not be followed by a homily.
If the Tourist Industry desired the service to be padded out to a greater length and dignified with rather more 'heritage', this could be done by a reaffirmation and confirmation (with great solemnity) of the anathemas pronounced by the Sixth Ecumenical Council against Pope Honorius I.