How can 6.10 ₿ be the average value of a Bitcoin transaction (lately)?
This is normal. Here you can see a chart of average transacted value every day: https://blockchair.com/bitcoin/charts/average-transaction-amount-btc
There are 2 things that contribute to this:

The very large transactions skewing the numbers. This becomes more apparent if you take a look at the median transaction values, which are around the 0.02-0.03 BTC range, closer to your expected value: https://blockchair.com/bitcoin/charts/median-transaction-amount-btc

The fact that many transactions aren't just sending money from one person to another person, but in most cases, includes change (sending money back to the sender's own wallet), and batched transactions (a single transaction that sends money to multiple people, often used by exchanges to process withdrawals while saving money on fees).