You haven’t actually said what you want to accomplish with the switch, port expander with a unmanaged switch works. But only if you only use a dumb network config in the router in the first place. Everyone on the same network VLAN and no need for PoE, no functions what so ever.
I believe that it isn’t what you expect from the switch based on the actual question that you have exhausted the router ports in the first place.
What is your future demands?
It is not that big of a price difference between a unmanaged switch and a managed switch that it will be of any meaning of buying the unmanaged switch. But the unmanaged switch is useless and worthless in the future if you change your equipment demands.
This will work because it is DHCP functions in the router and MAC address connected to IP addresses.
But the router will see everything like it would be connected to the router port that the unmanaged switch is connected to, the router will not see a switch! So the unmanaged switch itself doesn’t have a IP address!
A unmanaged switch is actually a raw muliplexer/demultiplexer.
The thing is that I made this mistake at first to buy a unmanaged switch. It worked as you want right now for about 6months before I bought a smart managed switch with PoE instead since I wanted a access point.
The factory quick start guide for a managed or unmanaged switch is the same thing. The only thing that differs is that you probably need to connect to the managed switch at start up to change static IP to DHCP managed from the router.
After that the managed switch works like a unmanaged switch with VLAN1 for all ports.
But if you change your mind in the future you have great opportunities with a managed switch.
A unmanaged switch ends up in the garbage if you change your mind.