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Joomla Core tables  utfmb3_general_ci

Yup. That's your problem. UTF8MB3 is the same as the plain old UTF-8 encoding, with up to three bytes per character. Some four-byte characters such as Emoji (e.g. 🏳️), Chinese characters etc cannot be encoded and cause a database error.

Should I update the entire database tables and columns to use the same collation? I assume utf8mb4_unicode_ci would be best?

Yes. This is the recommended encoding since Joomla 3.7. I had co-authored the code which made that possible in Joomla, with the all important finishing touches added by Richard Fath.

Are there any "dangers" of making this change to my system? E.g can existing records in the system become corrupted or inaccessible in a different collation?

No, nothing of the short. When you are going to a "wider" encoding (one which can encode more characters) you are not going to lose data. Only going to a "narrower" encoding (e.g. from four byte UTF8MB4 to  three byte UTF-8) you will lose data since the characters which cannot be encoded in the new encoding will be converted to question marks.

The other question is, of course, the PHP and Joomla version. That said, PHP version 5.4 and later do support the full Unicode charset internally, so they won't have a problem. Joomla 3.7 and later do support UTF8MB4 so, again, not a problem there.

You're welcome!