The dissolution of the HRE in 1806 came as a shock to many Central Europeans. Not only was Emperor Francis II's order illegal under Reich law, but there were a number of contemporaries that believed that the French invasions were an opportunity for the HRE to finally reform itself. Some expected the Habsburgs to lead the charge, but a few hoped that Napoleon would claim the HRE title for himself. The common denominator in these scenarios was that the creaky quasi-federal structure of the HRE could be reformed and made functional. The problem was that none of the major powers involved in Central Europe wanted such a revivified HRE. Napoleon opted to increase the strength of "third Germany" so they would be strong enough to serve as satellites but not independent enough to betray France. Both Prussia and Austria were much more concerned about their own houses in the face of the French threat than investing in an entity that strengthened smaller German states. The middling German states likewise preferred the arrangements of the Rheinbund which not only increased monarchical power, but also allowed for the takeover of mediatized territories. The HRE's patchwork of administrative structures and overlapping jurisdictions worked counter to these étatisme and imperial prerogatives. These conditions remained in place even after Napoleon's fall and do much to explain why there was no resurrection of the HRE.
Austria's Emperor and his chief minister Metternich had very little interest in the dead imperial title. The old Reich had evolved into a system by the eighteenth century that served to simultaneously limit the Habsburgs' powerin Central Europe while giving a degree of support to the Habsburgs outside of it. The Kreise and other HRE institutions constrained Vienna's hands in a crisis and added a further layer of diplomacy to gain the support of the German states. The HRE's mobilization system had been under increasing strain throughout the eighteenth century, but the Revolutionary and Napoleonic Wars underscored the importance of centralization and unified military command. The HRE's principles of collective defense was only barely sufficient for warfare under the ancien regime, but were patently insufficient for an era of mass mobilization. Moreover, the defection of the various middling German principalities like Bavaria to France further underscored that loyalty to the Habsburgs was not a given. Such defections did not come from the ether either; the rise of Prussia as well as the Bourbon interventions in Central Europe had shown the extent to which Central European loyalty to the Habsburgs was highly conditional.
Austria had a further reason not to push for a resuscitated HRE at the Vienna Congress. The expansion of the Habsburg patrimony into northern Italy and Galicia presented a set of problems for Metternich. There was a fear among certain policy circles in Vienna that the Italian nationalism stoked up by Napoleon would prove to be a Trojan horse working against Habsburg interests if left unchecked. It was much safer from Metternich's perspective to pursue a two-track policy with the German Confederation in Central Europe and a loose alliance system in Italy with no equivalent diet. Reviving the nominally supranational HRE meant potentially offering the Italian states a potential structure to work against Vienna's interests.
The German Confederation hashed out at the Vienna Congress was likewise amenable to a number of the German states. For one thing, it was a far more manageable entity than the HRE. The German Diet provided a degree of collective security for German states and a mechanism to air their grievances. Bavaria in particular was keen to ensure that the Confederation would allow for minor German sovereignty. The Confederation also validated the Napoleonic expansions of third Germany as well as the various regal titles. These titles were an up-jump in royal status and prestige than the various royals had enjoyed under the HRE. Only Hesse-Kassel remained a hold-out retaining its electorate title long after there was no more Emperors to elect. Prussia likewise recognized the potential strength of a smaller and leaner federal structure in Central Europe.
Turning the OP's question on its head a bit, the German Confederation performed many of the same functions of the late HRE while omitting some of the vices of the latter institution. The Confederation did uphold a degree of collective security against a potential foreign invader like France or Russia while not posing a potential threat to either Prussia or Austria. The German diet did allow for a degree of coordination over German issues among the German states but not in a manner that trampled over state sovereignty. Additionally, the lack of executive leadership like an Emperor was something of a virtue for many of the Confederation's members. No Holy Roman Emperor meant that the minor German states could not be dragged into foreign conflicts easily. this was not the case in the late HRE when the Habsburgs were both Austrian and HRE leaders and it was difficult to tell if fighting in a conflict was imperial in nature or was in the interests of the Habsburgs. And although the history of the German Confederation between 1815 and 1866 is one of Austro-Prussian rivalry, the Confederation was a robust enough structure to withstand several major crises without one rival coming out on top. Prussia's eventual victory in this struggle was a combination of unique political circumstances and adroit diplomacy in the 1860s which often obscure the many false starts towards Prussian-led unification.
The various supranational and federal structures of the HRE were a source of fascination for a number of Central Europeans after 1806. But this nostalgia for the Reich could not surmount the numerous obstacles in place that prevented its restoration after Napoleon. None of the policymakers at the Vienna Congress saw much need for an expansive confederated state with numerous guarantees for the rights of smaller powers. A neo-HRE would also open up thorny issues of imperial leadership as well as which states would be included in such an entity. None of the great powers' representatives at Vienna wished to go down such an avenue and without a great power patron, the idea of a restored HRE remained a pipe dream of a few intellectuals.
Sources
Forrest, Alan I., and Peter H. Wilson. The Bee and the Eagle: Napoleonic France and the End of the Holy Roman Empire, 1806. Basingstoke [England]: Palgrave Macmillan, 2009.
Vick, Brian E. The Congress of Vienna: Power and Politics After Napoleon. Cambridge: Harvard University Press, 2014.
Whaley, Joachim. Germany and the Holy Roman Empire. Volume 2, The peace of Westphalia to the dissolution of the Reich 1648-1806. Oxford: Oxford University Press, 2012.