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.

After some experimentation, and expanding on information in the comments:

Short answer

1. Because of bug.
2. But it doesn't really matter.

Long answer

[The problem]

This happens consistently in my Sanctuary settlement too, the only settlement where I have spotlights installed. I have two spotlight turrets; each is on level ground, and each is right next to its own dedicated generator.

The problem appears to be triggered by the passage of time rather than fast travel itself. For example, a very short fast travel (e.g., to Vault 111 and back) leaves the spotlights on, but a little further (e.g., to Starlight Drive In and back) and they'll turn off.

But most importantly, the problem is only cosmetic.

[The experiment]

Wait until a spotlight turret is in its darkened state, then:

1. Note that it's still moving; this strongly suggests that the game still considers it to be active.
2. Go into Workshop mode and notice your settlement's defense number.
3. Pick the turret up and set it back down. It should be lit now.
4. Notice your settlement's defense number is the same whether or not the turret is lit, so long as it's moving.

Then, to be extra sure:

1. Fast travel a good distance, for example, from Sanctuary to Starlight Drive In.
2. Return to your base. The spotlight should be dark.
3. Check your settlement's defense number. It's the same.

[So...]

While a little lighting in your settlement would be convenient—not to mention darned pretty—this appears to be a purely cosmetic bug.

You can sleep soundly knowing that those turrets, even when dark, are scaring away bad guys. And you can also sleep soundly because those turrets won't keep shining in your window. (Until Bethesda fixes this bug.)