I can answer the question in bold.

Technically speaking, no, there is no problem in having a six year old child bathe alone. He or she indeed can do it alone. And might enjoy solitary bathing, too.

But, bathing with one's mother is more fun because it is interactive and playful.

I have not had showers with six year old sons, but I have taken showers with six year old daughters (since you said that you would never shower with a 6 year old child, I thought that my experience showering with six year old daughters would be relevant enough), and I can tell you that the experience of showering with six year old daughters is wonderfully enriching for both mother and daughter.

It seems that getting a fun and wonderfully enriching experience should be justification enough for showering together.

Oh, and how do I know that the experience was not only fun and enriching for me (which would have been selfish), but also for the daughters? Because they laughed hard. I think that when a kid laughs hard, it shows that she is having genuine fun.

And, my then husband took our older daughter to bathe with him in a bathtub when she was a toddler, and I thought that it was absolutely great, too. So much fun for both of them. I took pictures of them together - it was so great that I did that, even though normally I was too lazy and stressed out to capture our day-to-day life in pictures.