Their closest political competitors that the Saud dynasty defeated in the 1924-25 war did claim the title of "Sharifian Caliphate" for themselves. Although Saudi Arabia is an absolute monarchy (e.g. there are no elections) and it is sometimes described as a "the Land of the Two Holy Mosques", which was seemingly the basis of the Sharifian claim to a Caliphate, the Saudis apparently avoided claiming that kind of (Caliphate) title. Is there a known ideological/theological reasons why the Saudis shied away from making such a claim?
The Caliph is the rightful religious and secular leader of the ummah, the community of muslims. As such, declaring yourself Caliph is at a minimum demanding token submission from all other muslim leaders; if not declaring them illegitimate and proclaiming rightful direct rule over their countries.
Saudi Arabia enjoys cordial diplomatic interactions with many other muslim nations, which is not really compatible with claiming to be a Caliphate.
Saudi Arabia in its infancy was a sheikdom client of the UK, the founder neither had the legacy nor legitimacy for such claims, in memoirs of those who worked with him, he was a very simple Bedouin with main goal to stay in power.
Also at that time, Saudi was very poor (oil has not been discovered yet), with uncertain future it will pass the rivalries among Arabs at that time, nationalism was on the rise and competition was tough among the new independent Arab states on who will be the lead country/ruler.
The corner stone of Saudi policy was being a client to the UK, and later on the USA.
With the increase of wealth came the Cold War, the rise of nationalism in Iraq/Syria/Egypt/Libya who allied with the USSR , while Saudi was with the western world. Saudi again was not in a position to be able to claim such a title.
The prestige of the title is not worth the cost, as it will make you responsible and in charge of Muslims worldwide, which will give you a lot of enemies right away. Plus a lot of discontent from Arab and Muslim countries, whom will resist having a higher authority than theirs.
The Saudis (especially MBS) generally don't like Islamists, such as Recep Tayyip Erdogan, who might do something similar if they were in charge of Saudi Arabia. "Caliphate" might also be considered a backward and uncivilized word by some investors in Saudi Arabia, as it was used by many early Islamic states.
The history and law is complicated, but there is a strong argument that one of the duties of a Caliph is to lead a military jihad against unbelievers and apostates.
The duty of Jihad was a collective one (fard al-kifaya). It was to be directed only by the caliph who might delay it when convenient, negotiating truces for up to ten years at a time.
So declaring yourself Caliph would be seen as a statement of intent to declare war on someone within the next 10 years.