Can I connect directly my MacBook Pro to NAS

Just to follow up - your network is pretty complicated, and I am having trouble visualizing the ethernet topology.

It'd be helpful if you could sketch out a rough diagram, showing just the router and switches.  Show how those switches are interconnected, the total port count on the switch, and how many ports are free.  Also show what room they are in.

Also, I'd like to review some basic principles (I apologize in advance if these are too basic)...

The ideal ethernet architecture for you would be to have one central switch that has enough ports to directly connect to every ethernet client you have.  This switch would be directly connected to one of your router LAN ports.  With this architecture, every pair of devices on your network can communicate at wire speed with no congestion.  That is because the backplane of the switch has enough capacity to carry that much traffic.  For instance, a 16 port gigabit switch will normally have a 32 gpbs backplane, so it can handle 1 gigabit per second in each direction on all ports simultaneously.

So over time you should aim to migrate your network set up to something reasonably close to this ideal.

There usually are some rooms where there are simply too many client devices for a single central switch to be practical.  In those rooms you deploy satellite switches so you can get enough client connections.  Those satellites connect directly to the central switch, and not to other satellites. 

The bandwidth between a satellite and the main switch is then shared by all the clients connected to the satellite.  The clients can connect to any device connected to the same satellite with no congestion, but there can be congestion when those clients are connecting to other local network devices.  So ideally these clients wouldn't all be in use simultaneously (or wouldn't need the full gigabit).  Note that 4K streaming uses 25 mbps or less - so streaming devices don't require as much bandwidth as many people think.

The central switch + room satellites design creates a "hub and spoke" topology.  Servers (including the NAS) should be connected directly to the central switch. If you are buying new switches, then a central switch with some multigig ports would be good.  Then the link to multigig satellites would be 2-10 gig, which would reduce the chance of congestion to/from the satellites.  Note the multigig links do need to be cat-6.

I'd need the topology diagram of your current setup to see how closely it matches the hub and spoke diagram architecture.  One thing I am thinking is that since you have two cables going to the game room, it might be possible to use one of those cables to connect the NAS directly to your router (or a switch connected to your router).  That of course assumes the unused cable is cat 5e.