The quest-giving source:
There should be a small radio receiver (handheld size) that the player can keep in their inventory under Misc or Apparel section. The player will need to activate the radio (on/off, or equip/unequip if it's an Apparel item) to receive Minutemen radiant quests in the form of emergency calls. These calls could be sent directly from the settler(s) needing help or by other Minutemen that are relaying the message across the Commonwealth.
Radio Freedom:
Radio Freedom is no longer used for calling for your help, but it should remain as an alternative channel to Diamond City Radio. It will report on general occurrences in the Commonwealth, but as a nice touch, it should have an extra focus on reporting of Minutemen and settlement related activity and events. For example, a feral ghoul threat that has been eliminated, a kidnapped settler rescued from Abernathy or County Crossing, or a Minutemen patrol that had successfully defended against a group of Gunners, and so on.
By shifting away from relying on Radio Freedom for the quests, players have the choice to listen to other radio stations or the ambient music (or no music at all) as they prefer.
Quest completion:
Not much to say here. The player should never need to report to Preston. The quest completes upon reporting to the settlers. Even better if the handheld radio is two-way, so the player can report from anywhere and at any time. The choice to report back in person should remain.
Preston Garvey:
Last but not least, Preston will no longer be a nuisance by virtue of a soldier commanding his General around. Preston will become purely a companion, not a lore/immersion-breaking quest giver. Players can now 'Talk' to him to follow the player, without enduring the annoyance of receiving three quests first. Fort Independence quest is an exception. Poor Preston gets a lot of hate and maybe this will alleviate that somewhat, if not completely.
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Of course, there is plenty of room for improvement where Minutemen gameplay is concerned, like delegation or even patrol route setups. This post is just to focus on how the radiant quests are given. It should be very simple to implement compared to other potential suggestions.
As much as I feel mods are essential, and playing on PC, this is something that should be how the game works as default i.e. the vanilla game. There's also the console market and I don't know how their modding scene will be in comparison.
Edit: Forgot to mention that I posted this elsewhere and got decent traction and suggestions to post it here where Bethesda apparently, on occasion, take up some of the suggestions to consideration. https://www.reddit.com/r/fo4/comments/3xz3mp/i_hope_bethesda_changes_minutemen_radiant_quests/