No. That's a situation where the lines blur between professional and personal. What you did seems reasonable

These things are very tricky; as there’s a fine line between showing concern and going too far.

Depends.. If the co-worker was male i would stay and offer more comfort and support.. If it was a woman i would restrict my forms of comfort to a verbal statement and then withdraw..

It has nothing to do with being warm or cold but being careful in a woke world of politically correct woman who can destroy you with one allegation of inappropriate touching.. Men have to be on guard around western woman these days..

Probably quite similarly. Those in grief certainly need comfort nd we should be forthcoming in that regard. But, I think, they also need some distance and solitude.

I wud just sit after offering water & let them clean their eyes.

You handled yourself professionally and in a caring manner.

You handled the situation perfectly.

What you did sounded like exactly the right amount.