Pretty much any of the events you see during the process of making a character fall in love with you. From the sweet awkward crushes to the first profession of love, to seeing them at the wedding, it's just really comforting watching 100% innocent romance in a game.
The festival dances in Harvest Moon 64. They're adorably bouncy.
The courtship of rival couples can also be rather sweet to watch - some notable ones in Animal Parade are Toby and Renee's engagement - they also appear to be at least one of the most harmonious couples in the game, not even bickering the way most of the others do in their previous scene - and Owen and Kathy's engagement, and both of Luke's scenes with Selena are priceless.
This can also happen in ways the game doesn't recognize with the way courtship works. In my case it was delivering the same gift to a person every day to try and bring their Relationship Values up. It felt like gaming the system until I realized my little farmer avatar was going out of his way to bring the girl he fancied a mug of hot, fresh-from-the-cow milk every morning come rain, hail or shine. Manipulation of game mechanics never felt so sweet.
When you marry Alisa, not being able to have a child with her is actually very sweet. Your character is basically confirming to Alisa you are marrying her because you love her and respect her - which means respecting her desire to remain a virgin (which is the implications of why she doesn’t have a child).
The proposals also bear mentioning on their own. Seeing the way the girls' faces light up at the man they love asking them to spend the rest of their lives with him, to the guys' embarrassment at the woman who has been wooing them being one step ahead of them. There's also the excitement on the faces of your betrothed's family and friends when you give them the news.
On that note, the weddings. Particularly the one between your character and chose love-interest. There's just something really sweet about seeing your love-interest all dressed upnote unless you married a secret bachelor/bachelorette, in which case they wear their normal clothes, and then telling you they'll make you happy, they love you, you'll always be together...
Watching your child grow in any game is absolutely adorable, and any cutscenes involving your child are even moreso.