There's no real sign that Taifa rulers really searched for a restoration of the emirate of Cordoba at their benefit even if they went trough regional unification by the middle of the XIth century.
They were more searching regional independence, not unlike what their precedessors attempted in the IXth/Xth centuries (which was as troubled for Umayyads than it was for Carolingians)
The strategical weaknesses of taifas were the same than al-Andalus during Umayyads, as in an extreme dependence on mercenaryship or recruitment from Christian Spain and Maghrib. While Umayyad Spain could, trough clientelism and sheer military strength, control their immediate neighborhood, there wasn't one taifa really able to do so, meaning the various Berber tribes were free to undergo their traditional cycle of unification/split/unification whom obvious targets were Ifriqiya and al-Andalus, further destabilizing these regions (and in the case of Spain, at the benefit of Christians).
Politically, Taifa's support essentially came from notable elites which weren't really thrilled at the perspective of a strong and intrusive power in al-Andalus, especially if the emir wannabee lacked a strong legitimacy (dynastical, championing Arabity as Umayyads did, and most of all beating up Christian kingdoms), or attempted to raise taxes.
Assuming Sabur al-Amiri does manages to make a bid for Cordoba (and abandon local leadership to his co-ruler which is likely IMO), defeat local opposition and survives (literally so, he would like a Clue character) the unmentionnable mess of what was Cordoba at this point, torn into multiple military-political factions, I'd rather see him restoring Umayyad dynasty under his influence.
How much would it be successful is anyone's guess, but its biggest impact would be allowing Umayyads to survive longer albeit without much real power and probably without great geopolitical impact apart a "Caliphal taifa" that would be an ambitious' magnet. Taifas would probably pay lip-service to the Caliph (especially if they make significant bid to control him) which would prevent neither the permanence of fragmentation or Christian influence and tributarisation of several taifas.
Still, the maintain of a unifying symbol could have some political consequences, but the growing illegitimacy of a Caliph unable to strike at Christians (which was really a determining factor of legitimacy in al-Andalus) would at best require from local "sultan" to sort it out himself and likely fail, if not for symbolical success, for the aformentioned reasons.
Of course survival of Umayyads is a "best-case scenario", and there's no telling when a local dynasty wouldn't pull an Hammudid, which would finally mercykill any semblance of caliphal authority in Spain.
