MacBook Pro 2013 doesn't connect to my home router through Wi-Fi

From the clear tests you ran thus far, it is clear you dont have an hardware problem and not any NVRAM issue.

Your problem looks like a specific disagreement on the way your router and your Mac manage the attribution of an IP to your Mac to start communicating ( the protocol used for this address attribution is DHCP ).

You can trouble check this disagreement by looking on the 2 sides of the required agreement.

On the Mac side

Open:

 > System Preferences... > Network

then select Wi-Fi on the left. You might see that your IP address is something like:

169.254.xxx.yyy

where xxx and yyy are numbers included in [0 - 255]. ( See APIPA address ).

Next unlock the paddle, click on Advanced... and select Hardware and check your hardware address. This one should like:

xR:xx:xx:xx:xx:xx

where xR and xx are 2-digit hexadecimal numbers ( in [00 - ff] ). If the R digit is:

2, 6, a or e

this means that your MAC address is a random one which will change at any new connection attempt. And the core of the problem is that this is an abnormal behaviour in a normal network ( this is the normal behaviour of a security tester or a criminal which would like to enter a target network with a stolen MAC address ).

Most routers configured to put on a minimal security won't accept such a rotating MAC address for a uniq machine, simply because this is not normal user behaviour.

But some recent recent versions of iOS ( 14 ) and perhaps undocumented updates of MacOS ( more technical investigation needed here ) are starting to use this MAC address randomization function to protect users privacy in an hot spot Wi-Fi network. But this approach is a total failure and is breaking the normal use of DHCP.

[ This caused a huge worldwide surge in support call to fix Wi-Fi network failing after recent mobile phones or computers OS upgrades. ]

On the router side

Check your router configuration which you can probably access through its router address. You will find this router address within the submenu:

Advanced... > TCP/IP > Router:

and search for the specific DHCP configuration tuning.

[ If you want specific help to further trouble shoot this side of the problem, please provide the brand of your router and a screen capture of this configuration menu. ]