Why didn't warmongers/fame-mongers like Cleon and Alcibiades suggest to attack the heart of Lacedaemon directly instead of this risky Sicilian expedition?

This is a really interesting question and not one that's easy to answer in a few words. Just for the sake of clarity: Kleon died at the battle of Amphipolis in 422 BC and had nothing to do with the Sicilian Expedition. Indeed, the Athenian defeat at Amphipolis and the deaths of Kleon and Brasidas led directly to the negotiations resulting in the Peace of Nikias (421 BC), which was still nominally in force at the launching of the Sicilian Expedition (415 BC). In other words, the Athenians didn't really choose to attack Syracuse instead of Sparta. They were formally at peace with Sparta at the time. It was only their repeated provocations (including raids on Lakonia by Athenian forces on their way to Sicily) that prompted Sparta to resume the war in 414/3 BC.

Still, it is an undeniable fact that the Athenians at no point in the Peloponnesian War seem to have entertained the idea of attacking Sparta itself. Despite numerous raids of the Lakonian coast, and the establishment of permanent fortresses at Pylos and on Kythera in 425 BC, they never invaded the region in force, and no one seems to have suggested that they should. Why not?

Older scholarship explained the lack of a strategy of annihilation on either side of the Peloponnesian War as a supposed feature of Greek warfare in general. It was argued that Greek communities simply didn't intend to destroy each other, which would have been an act of fratricide and sacrilege. In this view, wars were fought over small gains in land or plunder, and the communities involved were never really at risk. Invading armies would always stop short of actually attacking and razing the enemy's main urban centre.

The actual record of the Peloponnesian War, however, shows that this explanation cannot be right. In the course of the war, the Athenians and Spartans annihilated the settlements of Aegina, Plataia, Skione, Melos, and others. Thucydides records a lengthy debate at Athens over whether rebellious Mytilene ought to be destroyed completely, or whether only the thousand most prominent citizens ought to be killed. These atrocities were not a new feature of a more brutal form of war, either; the wholesale destruction of states and settlements is just as frequently attested in the centuries that preceded the Peloponnesian War. Clearly, the complete eradication of enemy communities was a real option. Yet Athens never attacked Sparta itself, and when Athens finally fell in 404 BC, Sparta decided to spare it.

J.E. Lendon in his book Song of Wrath (2010) explains this apparent lenience in a different way. He argues that Athens and Sparta were never intent on destroying each other, not because of moral compunctions, but because those were not the stakes over which they fought. The prize Athens wanted was supreme status in the eyes of the Greeks; Sparta resisted because Athens' prominence would go at the expense of their own. Since the war was fundamentally about recognition of a claim to leadership status among the Greek states, the complete destruction of the enemy would be counterproductive. The aim was, rather, to get them to admit the political reality in which they lived. Athens wanted Sparta to acknowledge the new supremacy of Athens. Sparta wanted Athens to remember its place, and pay homage to Sparta as the ancestral leader of the Greeks. The aim of the war was not for the enemy to be wiped out, but merely humbled, and forced to admit that it was less able than its enemy to protect the interests of others. This symbolic and ideological purpose naturally made the Peloponnesian War into a collection of proxy fights in which the main players themselves were never really at risk. It was only during the final phase of the war - the so-called Dekeleian War (413-404 BC) - that Sparta went for the jugular, and this was to a large extent because Athens no longer had the strength to do the same to them.

But there were practical reasons too. Sparta may not have had walls, but it was not undefended. Over land, it was ringed by mountains, which in turn were ringed by allies; there was no way to approach Sparta itself unless those allies could be brought to turn their backs on their hegemon. The Athenians would have to successively defeat or persuade first Megara, then Corinth, then Mantineia, then Tegea, and the Spartans themselves would have mustered an army to meet them long before they could reach Lakonia itself. The Spartans rightly saw the alliance of Peloponnesian states that formed around Argos in 420 BC as a far greater threat to their security than Athens ever was, and Alkibiades' efforts to support this anti-Spartan alliance (in violation of the terms of the Peace of Nikias) suggest that he may have seen its rise as an opportunity to destroy Sparta once and for all. However, this hope was crushed when Sparta brought its full might to bear on the Argive alliance at the battle of Mantineia in 418 BC, won a resounding victory, and regained its supremacy over the Peloponnese. Without Peloponnesian allies, the Athenians would have had no hope of approaching Sparta by land. (It is worth noting that the eventual Theban invasions of Lakonia in 370, 369 and 362 BC were made possible by, firstly, a larger army than Athens could have hoped to muster alone, and secondly, the active support of the majority of Sparta's former Peloponnesian allies. They still failed to take Sparta itself.)

It would seem obvious for the Athenians to stage a surprise invasion by sea instead. This is how they had traditionally attacked Spartan territory, bringing large fleets to raid the coastline, burn the dockyards, and attack outlying settlements. Their fortress on Kythera right across the sea from Lakonia was ideally situated to serve as a launch pad for a major invasion, and the Spartans would have almost no time to call on their allies and prepare. However, it was one thing to launch a raid on the coast, and quite another to mount a full-scale invasion on the scale necessary to give the Athenians a chance in battle against the full Spartan army, even if the Spartans were forced to fight alone. Naval raids were always based on hit-and-run tactics. Small detachments of heavy infantry would land and attack opportune targets, but run back to their ships if they encountered organised resistance. The ships allowed them not only to arrive in vulnerable places, but also to escape from there. But could such a method be employed with the full (post-plague) Athenian field army of some 10,000 hoplites, plus thousands of light-armed troops? If the battle were to end in defeat, would any Athenian have a hope of making it out alive? The Athenians mounted regular invasions of neighbouring Megara with their full levy, but this was largely because they knew the Megarians could do nothing to stop them. Against Sparta, the likelihood of pitched battle had to be acknowledged, and the option of defeat considered.

Indeed, the larger the army the Athenians sent, the greater the Spartan advantage. Due to their lack of training and rudimentary organisation, large Greek armies were cumbersome and vulnerable to panic. The Spartans showed at Mantineia that they possessed a great deal more tactical control than the average Greek army. Battle was an uncomfortably risky business in any case, but to provoke a tactically superior enemy to an all-or-nothing engagement in his own home ground would have been a deeply questionable decision, and to commit the entire citizen body to such a fight would be totally irresponsible. Add to this the fact that there was no retreat from Lakonia except by sea, and the average Athenian would likely have regarded this strategy as walking blindly into a trap.

The only realistic approach would have been the one that the Athenians chose in their attempt to conquer Sicily. They would invade with a relatively modest force, composed of equal Athenian and allied contingents, and seek to establish themselves in enemy territory, searching for an opportune target. This would minimise the risk to the Athenian citizen body in a single all-out engagement. But they could not count on winning such an engagement, because unlike the Syracusans, the Spartans were experienced in war; even numbers were not going to cut it, and the Athenians would need a force of significantly more than the 5,000 hoplites they initially committed to the Sicilian Expedition. And this force could not afford to linger, because with every day that passed, the Spartan army would be bolstered by its allies arriving from across the Peloponnese. And even if runaway helots were arriving at Pylos all the time, the slave population at large seems to have been happy to support Spartan rule, mustering 6,000 volunteers at the call of Agesilaos in 362 BC.

In other words, a single, decisive battle would have to be fought immediately, before helots and allies could flesh out Spartiate numbers and bring one of the greatest regional populations of Greece to bear on the invader. It would have to be fought either on the southern plains of the Eurotas, or at the towns of Sparta itself - where narrow streets proved easy to block for the Spartans resisting the attacks of Epameinondas in the 360s BC. In this engagement the Athenians would have no advantage except the mixed blessing of greater numbers. It seems most likely that such a battle would end in a massacre, and one that any rational observer would be able to predict - a more risky undertaking even than the attempt to conquer Sicily.