Mac absolutely cannot boot into recovery mode, reset NVRAM or access the boot menu.

So this 2011 laptop can still boot normally? It is only when you attempt to boot into Recovery Mode that you receive the circle with the line through it?

Why do you want/need to boot into Recovery Mode? Perhaps the same issue is causing both problems.

Try booting into Internet Recovery Mode using Command + Option + R assuming you are running macOS 10.12.4+ so that you can access the online macOS 10.13 installer.

The circle with line through it indicates a bootable macOS is seen, but it is not compatible with your Mac perhaps because it is damaged. If you are booting into local Recovery Mode using Command + R, then you are booting to those same damaged files.

Otherwise if this laptop still boots normally, then I highly recommend you create a bootable macOS 10.13 USB installer using the instructions in this Apple article while you still can:

Create a bootable installer for macOS - Apple Support

The 10.12 installer is broken when it comes to creating a bootable USB installer, but can be fixed using the information here:

Instructions to fix the macOS 10.12 Sierra installer

However, unless you have a third party app which is not compatible with 10.13 High Sierra, then I would recommend using High Sierra instead since that is the last OS compatible with a 2011 Mac.

Why do you want/need to boot into Recovery Mode? Perhaps the same issue causing you problems in macOS is also causing the problem booting into recovery mode.

If you are not having any problems booting into macOS, then maybe the issue is only with the built-in keyboard or perhaps even the Logic Board. I have discovered a couple Macs where the keys don't always work with the system firmware, but work fine once booted into macOS (sometimes even the other way around, although both are very rare occurrences).

Does the same issue occur when attempting Internet Recovery Mode using Command + Option + R, or even just a simple Option Boot when using a wired external USB keyboard?

Have you tried using the other Option key to try Option Booting?

You can use an external USB keyboard to access the various startup modes and perform a PRAM Reset. Only the SMC Reset requires the use of the internal keyboard.

Maybe force it to Option boot???

Make an Intel Mac Boot Directly to Startup Manager…

https://osxdaily.com/2021/02/26/make-intel-mac-boot-startup-manager/#:~:text=As%20with%20all%20command%20line%20activity%2C%20start%20by,sudo%20you%E2%80%99ll%20have%20to%20enter%20the%20admin%20password.?msclkid=19cc70e0ab9d11ecb47c8ddb8bcaef0f

Make an Intel Mac Boot Directly to Startup Manager (osxdaily.com)