You can do the flourishes and it says later you can ALSO interact for free. That would indicate the flourishes are not part of the interaction and therefore would not take the free item interaction allowing you to perform the flourish for free and then interact with an item for free as well.
Your turn can include a variety of flourishes that require neither
your action nor your move. [...]
You can also interact with one object or feature of the environment
for free, during either your move or your action.
If that was not the case they would have said that you can do a flourish or interact with an object for free once per turn to indicate you could only do 1 of those each turn and you had to choose.
That being said, in the end, it is always the choice of the DM (I remember seeing somewhere in the official rules that basically stated the dm can choose to customize/change rules as they see fit).
If I was a DM and a player was throwing a fit because I did not allow them to do this all for free I would give in but have something happen as soon as they set the item down (perhaps a slight earthquake, an animal, an npc, or a trap but something) and this thing would either move the weapon away or steal it outright (possibly making the player lose the weapon for good).
Do that periodically and it would quickly teach them a lesson of why arguing with the DM may not always be the best idea.