Recently, I saw someone arguing for a long time that "on-chain data" is solid evidence.

I find it a bit funny but also a little alarming…
To be honest, what you see on the chain isn't necessarily the chain itself, but the nodes you're connected to, the RPC (Remote Procedure Call) interfaces you're using, plus how the indexers organize the data.
If any part is slow or misses something, your "real-time" data becomes just playback.
Especially since some data tools or tagging systems are questioned for being laggy or even misleading, I’m not really surprised: tags are inherently based on human assumptions, and updates can’t keep pace with every transaction.
Anyway, now I tend to cross-check data from two companies, and for important info, I prefer to look it up myself in the block explorer—better to be a bit more trouble than to be misled by a pretty dashboard.