In the beginning of the game, I encountered several small story events, like Joja giving away coupons or encountering the hermit at night.

But since about one of the patches hit, there haven't been any of these small events (the ones with skip buttons available on). It's not like I feel I'm super far into the game. The last of these small events I've had were before the flower festival in spring (year 1), I'm now well into summer with no sign of them.

Can anyone shed light on how these things trigger, and if something should've been triggered (I think I read somewhere advancing friendship levels might trigger them, at which point could I go for to test?) if there's a way to restore them without starting over?

Most events are triggered at every two hearts of friendship (2, 4, 6, 8, 10) at least that's how the events with single people happen.
Events are place specific and usually time specific, for instance one character will have an event based on going to the beach before 12 PM in Summer.
Some events start on a new day when you exit your house as well.

I've gone a full season without events and had a season with several just because I had been raising heart levels with various villagers.

The best advice I can give is to check locations where a villager regularly visits after raising their heart level to an even number. And check several times a day if you really want an event.

I've found that a lot of events tend to happen naturally as long as I'm exploring several different areas a day rather than just farming then mining. Visiting homes tends to get me the most.
Take time to walk through town instead of around it. Go into people's houses daily, particularly if you've raised their heart levels.
I also think there was at least one event I witnessed that required a certain heart level for all family members as well. I could be wrong, but I only got the event after raising all family members heart level to two.

This is just what I've experienced, though.

EDIT: And as for missing the events, I had one event that was supposed to trigger at 2 hearts with one of single characters trigger after seeing the event for 4 hearts (according to the wiki), so I don't think they're limited to happen only at a certain level and are lost after raising friendship beyond that level, just that they're limited to time and place.
If you still can't trigger any (check the wiki for a list of events if you want to make sure you're seeing them, they're listed on individual villager's pages), you might have a bug and you'd be better off making a report on the official forum where someone who works on the game or is connected to ConcernedApe/Chucklefish might see it.

Very sound advice. But not really being the kind of guy who goes WIki a game before he gets stuck, I didn't even think to check the wiki for events.

I gifted Leah a flower, so the level was now at 1 heart + 2 checks in gifts. The relationship stayed like that for about 3 days where I ran into her on several occasions where I couldn't talk to her at all.

Once I quit the game, next I started it I was at 2 hearts 0 gift checks. I figured there was supposed to be an event for that transition which didn't happen the 3 days I'd spent. But reading the wiki, I can see I needed to go to her house which has so far been locked to me. Guess it shows what I know :-)

I were just feeling kinda upset if I had to restart my 11 hour game to get story snippets. Thank you very much :-)

Sometimes the wiki is the only way to know for certain. I'm glad you worked it out, though. ^^

if someone still has similar questions, write to me in the PM , I will try to help you!

I wrote you in private messages!