I'm not a medievalist, but the readings I've done have left me with the impression that during the High Middle Ages, the thing that distinguished kings from the princes and magnates of their realms was less their power and authority, and more their sacral role.
John France discusses the nature of governance in Western Europe during the Age of the Crusades
Landownership was the distinguishing characteristic of the ruling elites of western Europe. In itself it conferred power, as it always had. Kings claimed an overarching sovereignty and proclaimed that others exercised governmental power only as their delegates. But in practice they were not sharply distinguished from other landowners, who shared their authority rather than acted as its agents. Nor was the pattern of royal ownership in any way different from that of other powerful men and women. [...]
Throughout our period the primary form of wealth was land, and everywhere its ownership on any scale gave quasi-governmental powers. Landed property was seen in a very absolute sense and its very concrete nature tended to overshadow sovereignty, for power sprang from possession of land. To be a king was of necessity to be first landlord of the realm, a position which merged into that of other landlords and blurred the distinction between the king and others, and that between sovereignty and ownership. [...]
In fact, there was a deep confusion of what for us are two separate concepts – sovereignty and ownership. Every kingdom was a lordship and every king had the preoccupations of a landowner: every lordship resembled a kingdom and its holders felt entitled to some of the perquisites of a king.
To what extent did this also hold true for the earlier Carolingian kingdom? Did the magnates exercise much autonomous authority in their capacity as magnates, or was the whole realm run by the royal bureaucracy?
(Expanding on this previous post)
The reality of public service in Francia essentially crumbled along Merovingian collapse during the late VIIth century.
Whereas public service was more or less maintained under the strong rule of Clothar II and Dagobert I, the decline of royal authority led to a slow transformation of palatine service with tendencies of autonomisation (control over public fisc being a source of personal power, for example), regionalisations (where sub-kingdoms each had their own palatial administration even under a same king) and functional systematisation where duties and jurisdictions were carefully precised in a context of numerical devellopment of a renewed aristocracy.
Ineed, the royal faida seeimingly caused the extinction of several important families, traditionally associated with public service, and the rise of a more militarized and more regionally-minded than their predecessors : altough the kings could bypass what had technically became a law not to appoint public servants outside their region of origin, more and more of these became territorialized and entranched themselves in their jurisdiction in no small part thanks to the maintainance or creations of religious establishments.
The crisis of royal authority (attributable mostly to successive royal minorities and the loss of hegemony outside Gaul) gradually reinforced this transformation, with tendencies to irrevocability and hereditarization. This provoked several conflcits among the aristocracy : a family or a set of families that threatened to control too much of public power locally was fought against by others (with royal arbitrage), but these often ambitioned to do that for themselves. This was both true of ecclesiastical and, maybe modelled from this first situation, lay power.
This strained the capacities of the royal fisc, with endless donations to this new forms of aristocratic powers and to the church, while confiscating properties became almost impossible (less in defiance to royal power itself, and more because it would have benefited whoever dominated the royal council) with charges and revenues being diretly payed and extracted by local powers.
Eventually, it meant public service was in constant rivality with aristocratic power in Francia proper which while relatively autonomous from royal authority (or being identified to it) was too tied to their own interests and clientele to be entierely considered as public service anymore. This led whole regions to become essentially autonomous, if not independent : Aquitaine, Provence, Alsace (with the exemple of Adalric-Etichon in this recent post), even Maine; while peripheral dukes/petty-kings effectively removed Merovingian dominance into turning their duchies into independent polities.
The rise of Peppinids didn't really help, in the sense they promoted an "alternative hierarchy" based on vassality and relatively exclusive relations with their own lineage. It's not so much they directly attacked a post-bureaucratic model than they made it redundant and was more political efficient as involving a direct and stabler aristocratic obedience that the palatial network (they controlled in Austrasia) couldn't bring anymore.
It was still an important ideological model in the VIIIth, the structures of the kingdom being tought in the idea of public service nonetheless; and Peppinids-Carolingians tried to re-establish it : by crushing peripheral principalties one by one, by promoting "affiliated" religious establishment and bullying autonomous or rebel ones thanks to an alliance with Papacy, and by enforcing an institutional hierarchy at their benefit and on their dependence.
Promoting a rebirth of public service along the old Frankish model, however, wasn't really possible anymore even by bullying resisting aristocracy. If the re-hierarchisation and direct interventionism of the Carolingian was a success, it didn't removed either territorialization or perenisation of local elites, just made them more susceptible of intervention. By reframing a systematized vassalic network into terms of late ancient service, Carolingians effectively concretized their fusion : counts, abbots, vassi were effectively all associated with militia, its privileges and implications of delegations and benefits from the charges; while legitimazing territorialisation and hereditarisation of nobility.
It had benefits : the use of precarii, or ecclesiastical/monastic lands included into state service being lent to lay beneficiaries (which was a really efficient way to "money laundering" confiscated or ill-gained goods) removed a good deal of fiscal pressure on Carolingians, for instance, and the inclusion of clergyen and bishops as such into palatial service provided with a safe recruitment pool. As long precarii and conquests flowed in, the aristocracy was controlled by strong kings (the Frankish nobility seems to have been fairly limited to maybe an hundred of families at this point, the other falling prey to Peppinids-Carolingian takeover)
That said, Carolingians remained effectively an aristocratic family among others that "made it" and while sacralized by their alliance with the Papacy trough blessing and a relatively superficial imperial deonomination; didn't had the same transcendental nature than their predecessors that allowed them to navigate trough changes : confiscating and removing nobles even during the early Carolingians was easier said than done; missi dominici had little effective power and were more devices of negociation and scrutiny, etc.
It's one of the reasons bishops, altough not entierely apart, along with their intellectual and ideological support for imperial display (much more Frankish and Christian than "Roman") were one of the favourite institutional devices of the empire,with immunities (basically grant of territorial privileges) being generalized to both distach them from neighbouring families and attach them to palatial service.
Even if Peppin or Charlemagne would have wanted to, they couldn't go above negotiations and council with their nobility. Regularily during the general plaids or conventi generali which were effectively assemblies that influenced the make up of capitulari and edicts, while presided and made by the palatial service. Irregularily with the missi dominici as envoys to "problematic" nobles. Even the...complex matrimonial policy of Charlemagne is a testimony to the needs to coming to terms with such and such lineage, while Merovingians often elected at their apogee to marry outside the kingdom.
Trends of regionalisation themselves were still as strong as ever, leading to Charlemagne's succession being originally made along Francia/Aquitania/Italia without transmission of the imperial title, before every son besides Louis died, part of the cause of the following civil wars being the tentative of Louis and Lothar to force against regional tendencies.
Did the magnates exercise much autonomous authority in their capacity as magnates, or was the whole realm run by the royal bureaucracy?
"Both and neither" is a frustrating answer, but the early Peppinid-Carolingian Francia does looks as it belonged to both Late Antiquity while announciating changes of the Xth-XIth centuries. The whole public service apparatus was mostly a remodelling on the new political situation and even if Merovingians already more readily accompanied changes than fought them, there was little being strictly bureaucratic now. Rather regional nobles owning fidelity and service exclusively to the new dynasty : the collapse of Carolingia point to the fragility of what was effectively a weak state (weaker than Merovingian Francia, one could say) where direct negociation between aristocracy and the palace could be held as long you had strong kings.
You could say that what distinguished Carolingians in their hey day from the rest of imperial aristocracy was their capacity to what remained of public institutions to reinforce their own aristocratic network and systematize it in all the Empire after having dusted it off. It was really dependent on continuously held prestige and power, and if Carolingians kept for almost two centuries enough prestige and sacrality, it didn't really save what early Carolingians attempted to build-up.