Yes, but there’s more.
You are correct in assuming that an archer firing into an area of heavy obscurement has a normal attack roll due to advantage imposed by the target not seeing them and disadvantage imposed by them not seeing the target.
However, the part between your unseen attacks quote is also relevant to the attack. The question you ask in your post is also different than the question in your title. You asked additionally, “Does the spell do precisely nothing in terms of making us harder to hit?”
And so I quote the part you left out:
PHB Pages 194-195. This is true whether you’re guessing the target’s location or you’re targeting a creature you can hear but not see. If the target isn't in the location you targeted, you automatically miss, but the DM typically just says that the attack missed, not whether you guessed the target's location correctly.
The DM can determine whether the attack hits or misses without referring to the attack roll in certain circumstances. For example, if you cast the Fog Cloud spell and then use your movement to move out of the cloud or elsewhere within the cloud, but still remain mutually obscured to the enemy, the enemy would have to be able to hear you to get a normal attack roll.
If it’s ruled that the enemy can’t hear you, perhaps by a low perception check roll, passive perception at the DM’s discretion, or the deafness condition, then the attacker would have to guess your location. The attack could then automatically miss regardless of any attack roll up to the discretion of the DM. In the event your location is guessed correctly, however, the attack is made without either advantage nor disadvantage.
So, to answer your second question: no, the spell does not do absolutely nothing to make you harder to hit. You should also consider the case where the enemies have advantage before the Fog Cloud spell, as well, and the effect that it has on cancelling out said advantage.
You can hide from the archers separately to the melee fighters
From PHB pg. 177:
Hiding
When you try to hide, make a Dexterity (Stealth) check. Until you are discovered or you stop hiding, that check’s total is contested by the Wisdom (Perception) check of any creature that actively searches for signs of your presence.
You can’t hide from a creature that can see you, and if you make noise (such as shouting a warning or knocking over a vase), you give away your position.
When you hide, you make a stealth roll. This is against the passive perception of the enemies you are hiding from. If some have higher passive perception than others, and your stealth roll happens to land in between, you will be hidden from those with lower passive perception but not hidden from those with higher passive perception.
Passive Perception. When you hide, there’s a chance someone will notice you even if they aren't searching. To determine whether such a creature notices you, the DM compares your Dexterity (Stealth) check with that creature’s passive Wisdom (Perception) score, which equals 10 + the creature’s Wisdom modifier, as well as any other bonuses or penalties. If the creature has advantage, add 5. For disadvantage, subtract 5.
So, this sets the precedent for being able to hide from some but not all enemies. If the archers cannot see you, you can hide from them, but the melee fighters can see you and thus you cannot hide from them. The fact that the melee fighters can see you does not stop you from attempting to hide from those who cannot see you.
Best Answer
Yes, that is the rules-as-written.
It's worth noting, additionally, that there's actually no need for the archer to step out of the fog cloud in most circumstances: despite being effectively blinded, per the rules of the game, a creature would still know the location of another creature they cannot see, so long as that creature does not take the Hide action and roll a Dexterity (Stealth) check high enough to beat the archer's passive Wisdom (Perception) score (subtracted by 5, due to disadvantage conferring a -5 penalty on passive checks). Once the targeted creature hides, it would only then become necessary for the Archer to step out of the fog cloud to find their target again—or they could simply take the Search action to make a Wisdom (Perception) check (with Disadvantage) to try to locate the creature from within the Fog Cloud.
But so long as the Archer knows the location of their target, then from Disadvantage (from not seeing the target and attacking at long range) and Advantage (from the target being unable to see the archer) the Archer would get normal attack rolls up to their maximum long-range distance.
Yes, which is why at my table, I use a houseruled "fix" to this specific issue, that in order for being "unseen" to confer advantage on your attack rolls, you must also be able to see the target. The general consequence of this change is that it causes attack rolls inside Fog Clouds, Darkness, etc., to generally all have Disadvantage, as opposed to the RAW circumstances which would instead cause attacks to be made normally. To me, this seems like an appropriate fix because it properly causes blinded combat to diegetically feel clumsy and confusing.