No

You don't have to look after that beyond the first few missions for the Minutemen (you don't even have to do these, if you don't want to). The first few missions for Preston and Sturges will have you setting up power, water and food in Sanctuary and from there onwards settlements can be largely ignored and depending on how you play the game. There are spoilers below. I'm not marking them up and you have been warned.

But...

You'll get Sanctuary early on and meet up with Preston at Concord and this will fully establish a settlement with people in it. This settlement is the primary base of the Minutemen and will remain as such unless you progress the Minutemen quest line and retake the Castle, at which point this becomes the primary Minutemen settlement but the original survivors will remain in Sanctuary.

In addition to these two settlements, if you continue with the Minutemen you will frequently find yourself in a position where Preston wants to expand by helping another settlement, which will then join your cause and potentially get attacked, prompting you for assistance (which you can ignore if you don't mind them dying or if that settlement is properly secured).

If you do not progress the Minutemen quest line and instead go with the Institute, you won't end up getting any additional settlements throughout the game.

If you follow the Railroad, they'll ask you to take a settlement as their new safehouse after one of their existing safehouses is destroyed by the Institute. This settlement doesn't need a broadcast tower as part of the mission and therefore won't generate a population of settlers that need looking after.

If you follow the Brotherhood of Steel, the airport ends up becoming one of your settlements during their quest line.

The additional settlement (or in the instance of the Minutemen, a settlement of your choosing) is required in order to build the Signal Interceptor which is used to infiltrate the Institute later in the game.

I got sick of waiting for an answer to my broader question regarding how the damage multiplys in general, so I did a bunch of testing myself.

Method:

All tests were done by shooting a friendly setler in the back of the torso at close range.

The NPC was wearing nothing but a Minuteman outfit and had their HP set to 5000 with the console. I did not check what their Damage resistance was nor did I modify it, though I probably should have.

A single shot was taken, NPCs HP remaining checked with the console, then the game was reloaded. This was done 5 times.

Due to perks etc, my Sneak attack bonus was at 4.2 Not sure what my exact critical bonus is, I had 4 ranks of Tesla Science (+20% crit damage) and no other perks or mods which would increase it.

Tests performed with a Laser Rifle, instigating tests performed with Old Faithful with the same mods. Both showed a base damage of 144 in the Pip-Boy interface.

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Results:

Normal Shot

111.81, 214.99, 213.32, 214.57, 212.48

Average: 204.72 damage (Average may be low? First shot did much less damage)

Critical Hit:

350.93, 349.86, 350.63, 350.34, 351.00

Average: 350.552

Sneak Attack:

905.80, 905.97, 906.70, 906.38, 905.46

Average: 906.062

Sneak Attack + Critical Hit:

1040.25, 1042.20, 1040.83, 1038.45, 1040.27

Average: 1040.40

Normal Shot + Instigating Weapon

425.43, 430.58, 431.57, 430.70, 352.29

Average: 414.114 damage (Average may be low? Last shot did much less damage)

Critical Hit + Instigating Weapon:

565.44, 559.52, 566.39, 565.26, 566.12

Average: 564.546

Sneak Attack + Instigating Weapon:

1810.25, 1812.87, 1813.63, 1813.38, 1812.00

Average: 1812.426

Sneak Attack + Critical Hit + Instigating Weapon:

1944.69, 1946.42, 1947.35, 1947.74, 1947.11

Average: 1946.662

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Discussion:

The above results show some variance, with some shots noted as being much lower than others. It is not clear if this was due to poor accuracy causing some bullets to hit the wrong location (though injuries on target suggested all shots hit the back), natural RNG spread of damage, or damage resistance of the target.

When roughly comparing Sneak attacks performed with both Instigating and normal weapons, the stated Sneak Attack bonus does appear to be multiplicative with Instigating ( [Weapon Damage x2] x4.2) or at least close enough to satisfy me that this is the case.

Interestingly, Critical Hits do not appear to be multiplicative with Sneak and Instigating. In all situations the additional damage inflicted was close to the weapons stated base damage (144, give and take some variance). This suggests to me that critical strikes are seperate damage which is calculated and then added after sneak attacks and instigating are multiplied ( {[Weapon Damage x2] x4.2} + [Weapon Damage xCrit Modifier])

The Wiki suggests that Critical Hits use the following formula:

Ranged: Critical Hit Damage = DamagePaper + DamageBase x CriticalMultiplier

Melee/Unarmed: Critical Hit Damage = DamagePaper x 1.5 + DamageBase x CriticalMultiplier

DamageBase = Weapon damage, including workshop mods, but excluding perks and other bonuses. DamagePaper = Damage displayed in PipBoy (PipBoy cuts off decimal places)

Conclusion:

Sneak Attack and Instigating are multiplicative with each other.

Critical Hits are not multiplicative with Sneak or Instigating

This suggests: ([Weapon Damage x Instigating] x Sneak Modifier) + (Weapon Damage x Crit Modifier)