In Leviathan, c.13, Hobbes depicts the state of nature as 'a warre of every man, against every man' (Tuck: 88). (All references listed at end of Answer.)
Exactly what is the state of nature and what is so bad about the war that it involves?
The state of nature and its disadvantages
Here's Gregory Kavka's summary account (one of two Kavka quotations I'll use):
Hobbes argues that people living in a state of nature, without a common
power over them to keep them in awe, are in a state of war of every
person against every other. He defines war not in terms of actual fighting,
but as a known willingness to fight. So a war of all against all is a state
in which each knows that every other is willing to fight him, not one in
which each is constantly fighting. But it is more than just this, for Hobbes
contends that, in the state of war, there is so little security of life and
property, that all live in constant fear and productive work is pointless.
(And he uses this contention as the ground of a favorable comparison
of absolute sovereignty with the state of nature.) But, because people
learn by experience, a state of known universal willingness to fight would
not long leave its inhabitants feeling so insecure unless acts of violence,
coercion, and theft actually occurred with some frequency. Hence, the
real conclusion that Hobbes draws (and needs) is that the state of nature
is a state of war of all against all, punctuated by frequent violence, in
which the participants correctly perceive themselves to be in constant
danger. (Kavka: 292.)
So in the state of nature life is totally insecure and human flourishing is impossible because there is no safety in activity, production or possession.
How to avoid or end this state of affairs?
What are the options?
In Leviathan, Hobbes offers a hypothetical contract argument in
support of the conclusion that political sovereigns have absolute and
unlimited authority over their subjects, and that-save when their survival
is immediately at stake-these subjects are morally obligated to obey all
of their sovereign's commands. Hobbes, in effect, imagines rational self-
interested parties in a state of nature choosing among three alternatives:
(1) remaining fully in this state of nature; (2) grouping themselves together
under a government with limited, or divided, power and authority; and
(3) forming themselves into a civil society, or commonwealth, governed
by a sovereign with unlimited power and authority. He contends, however,
that the second alternative is basically illusory. Because of the constant
danger of factionalism, civil war, and social disintegration in a group
governed by a limited or divided power, such a form of social organization
does not provide its members with sufficient security to really remove
them from the state of nature.30 Each is forced, in order to protect his
long-term well-being, to anticipate-to use force and cunning to increase
his power and thus his ability to come out in a favorable position when
the inevitable civil strife breaks out into open warfare.3' The conflict of
each person with every other thus goes on, perhaps in muted form, with
trickery and deceit being more common forms of aggression than overt
violence, but nevertheless leaving each person in a basically insecure,
and thus unhappy, position.
Now as the formation of a government with limited or divided powers
would not constitute a real escape from the state of nature, the choice
of the parties, in Hobbes's hypothetical contract theory, effectively reduces
to one between absolute sovereignty and the state of nature. The argument
that the state of nature is a state of war of all against all supplies the
crucial remaining premise that allows Hobbes to conclude that the parties
would choose the absolute sovereign as the lesser evil. (Kavka: 296-7.)
A non sequitur
We ought to note at this point that Hobbes' argument at most supports the institution of a sovereign with, not absolute power and authority but merely sufficient power and authority to maintain civil peace.
But can we trust the sovereign - Leviathan - not to make things worse than, or as bad as, the state of nature?
Suppose we grant Hobbes his sovereign with absolute power and authority, what guarantee do we have that the sovereign will not treat his/ her subjects in ways that are as bad as remaining in the state of nature - or even worse?
Hobbes' reply would be that there is no state worse than the state of nature. Therefore by simple deduction the Hobbesian leviathan cannot be more fearful than the state of nature. Hence if the sovereign were to reduce us to conditions as bad as the state of nature, we should be no worse off than if we had remained in it. But Hobbes argues that it would not be in the sovereign's interest to reduce us to such conditions since 'in the dammmage, or weakening of [his/ her] Subjects' the sovereign's 'own strength and glory' would be fatally at risk (Leviathan, c.13; Tuck: 128-9.) In other words Hobbes believes that the sovereign has prudential reasons to create and maintain the conditions of civil peace, the components of which are defined by the nineteen laws of nature enumerated and specified in Leviathan, c. 15 - moral principles which when enforced are 'the way to peace' (Robertson: 144). Hobbes accepts, therefore, that even absolute sovereignty will dissolve under certain contingencies: 'the physical powers of a monarch or a sovereign assembly [a sovereign: GT] are never great enough, in themselves, to deter violent opposition' (Kavka: 308). Even ruthless power can evaporate under unremitting revolt.
As to the individual:
If the sovereign - as you say - 'can kill any individual in the name of authority', he or she cannot kill all; and any individual, no matter if justly condemned for an offence, has no obligation to comply with any commmand that aims at his 'hurt':
If the Soveraign command a man (though justly condemned,) to kill, wound or mayme himselfe; or not to resist those that assault him ; or to abstrain from the use of food, ayre, medicine, or any other thing, without which he cannot live; yet hath that man the Liberty to disobey' (Leviathan, c. 21; Tuck: 151).
The individual always has the chance (and with Hobbes' blessing, the rightful liberty) of escape, however tyrannical the sovereign may be.
I do not claim that Hobbes has demonstrated to all reasonable satisfaction that the rule of 'leviathan will be less fearful than the state of nature'. But I have assembled his reasons for supposing that it would be less fearful. They are worth consideration whatever one's final view of them.
Hobbes, Leviathan, ed. R. Tuck, Cambridge: CUP, 2018.
Gregory S. Kavka, 'Hobbes's War of All Against All', Ethics , Jan., 1983, Vol. 93, No. 2 (Jan., 1983), pp. 291-310.
G. Robertson, Hobbes, London: Blackwoood, 1886.