Helping Preston and forgetting about Shaun

I've tried to roleplay a character with that attitude before. Short answer is 'no'. The longer answer ahead contains many spoilers:

The objective of the Minutemen is to bring peace and order to the Commonwealth by defeating the Institute.

You can't destroy the Institute without entering it first.

You can't enter the Institute without defeating Kellog.

You can still try to chase down Kellog while assuming that Shaun is dead, as the game lets you respond that you're looking for your spouse's killer, rather than for your son. Most of the dialog choices support this. For instance, during Reunions you can tell Dogmeat to go find Shaun, or to go find Kellog. There are a few times when you are forced to indicate you are looking for Shaun, though. There's a point in Story of the Year, which is technically optional, where you have to indicate to Piper that you're in Diamond City to look for your son. There are also a couple points when talking to Valentine where you can't get out of it, such as when he's interviewing you after he's rescued.

At this point, other than making assumptions after leaving the Vault, the only real reason the player has to believe that Shaun might be alive is Mama Murphy. After you rescue Preston's group from the Museum of History, she indicates that Shaun is alive, and that you should look for him in Diamond City. Her knowledge of Sanctuary is some evidence she may be right, but her vision of you leaving cryo is significantly more convincing.

So, if you're trying to really roleplay this, I suggest focusing on the Minutemen, and not seeking out Valentine until you happen to be in Diamond City for some other reason. As long as you're there, you might as well check on Mama Murphy's vision. You can then rescue Valentine as part of your general Minuteman heroism, and things progress from there.

During Reunions, you get more evidence that Shaun is alive, culminating in Kellog directly telling you that he's in the Institute (assuming you let him talk), and his log entry that you need to read in order to continue the quest.

However, shortly after you enter the Institute, events can unfold in such a way as to allow you to resume believing that Shaun is dead (correctly).

Once you get into the Institute, wait until it becomes clear that young Shaun is a synth, then kill Father. After a quick interaction with young Shaun, Father comes out of the next room and issues the recall code. This makes it pretty clear that young Shaun is a synth. This would reinforce your character's initial conclusion that the child is dead, and also explain all the evidence that he might have still been alive. At this point, Father has already indicated that he's in charge of the Institute, the organization that killed your spouse, and apparently your child. It seems sufficiently in-character to me to just blast the guy at that point and be done with it. If you're having trouble initiating combat because you're in a conversation, just back up a little bit.

If your character isn't that violent, and you choose to listen to Father, he eventually claims that he is Shaun. You can still indicate that you don't believe him. You could end the conversation at this point and kill him, or simply refuse to cooperate at the end of the conversation.

This will result in you being kicked out of the Institute, at which point you can continue with the Minutemen quest line without ever worrying about Shaun again. This actually has the added benefit of expediting the Minutemen quest line, which requires that you eventually be kicked out of the Institute anyway.

It really depends on your definition of "The Minutemen Story." If you consider all main quests that the Minutemen can have, then no. You'll need to complete the game to do some main Minutemen quests.

For instance,

With Our Powers Combined is a Minutemen main quest in Fallout 4. It occurs after the ending if the player finishes The Nuclear Option and Old Guns and has (or gains) hostile status with the Brotherhood of Steel.