I think you're both right.
There's been a drop in popularity as newer technologies became available. Pixel art got as big as it was back then as it was due to limitation, they only really had pixel art to make games with. Computers and consoles back then could only handle so much. after 8-bit I think 16-bit came. Nowadays though there's a lot more you can do with pixel art even. Older games have a limited amount of space and a limited pallette but now you can add more colours and have more space provided you're up to the intensive task of making pixel assets. That leads to a lot more new styles for pixel art. And yes! Lots of indie-games use pixel art. I've seen a lot of new styles come up lately by indie game-makers. Indie-games got big lately. : D
Some people like the old-school pixel styles, others like modern pixel art. I guess it's still up to taste to a big extent though.
There's a charm to pixel art if done well.
Here's an example of pixel art from Digimon World Dawn. It's kinda VP save that it's a console game rather than a browser game but yeah your digimon are basically like pets that you go explore with and you get into fights with wild digimon.
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While I can't see 8-bit work well for a pet based or human avatar based site unless they're fine with being small, which is not bad at all if you're running it for pure fun.
Maybe something like the above could work for browser based pet games. Animated would be really cool but a still pixel image could also work. There could be customization if it was as big as the bigger sprites above, I think. If not a full blown game with exploration etc.. I think it could work for adopts..wait yeah now I remember Dragon Cave.
Probably difficult for a browser based VP but I though I'd show a screenie from a game called Owlboy. This is more modern pixel art.
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It's got a much smoother feel than most pixel work from before 2000 yet still retaining that pixel feel.
And one of the most beautiful pixel games imo I've ever played,
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Pokemon mystery dungeon sky. The characters stand out well against the softer environment.
There's lots of ways to go. If done well enough I think there is lots of potential for a pixel game to be popular among those who usually play games in other styles even in this day and age.
It's pretty nice how many different art styles and techniques exist. Pixel or otherwise.