This was it. I followed the instructions here https://stadicus.github.io/RaspiBolt/raspibolt_30_bitcoin.html
this time using the aarch64-Linux download of bitcoin core which I found at bitcoin.org/bin/bitcoin-core-0.20.1/

Any idea why this option isn't shown on the list of downloads at either bitcoin.org or bitcoincore.org? I'm also confused about why the arm-Linux version worked for other people doing installs on the Raspberry Pi. Maybe it works on Ubuntu Server but not Ubuntu Desktop? I'm not familiar enough with architecture differences to know.  

Thank you to all that helped me out.

I solved the issue with the hash. I was attempting to verify the unzipped file.

I did run:
sudo chmod +x bitcoind
This did not change anything regarding the errors from before. Still getting no such file or directory for /usr/local/bin/bitcoind

~/usr/local/bin/ was a mistake on my part. I've been using /usr/local/bin/ and can confirm I am running everything from root. This is all happening on a clean verified install of Ubuntu for Raspberry Pi.

Followed the instructions here
https://stadicus.github.io/RaspiBolt/raspibolt_30_bitcoin.html
Was able to get all the way back to where I was last night with the proper ARM version of Bitcoin core (and verify the download this time). I now have bitcoin-cli, bitcoind, etc. in the /usr/local/bin directory as outlined in the RaspiBolt instructions.
However when I run: bitcoind --version          
I get bash: /usr/local/bin/bitcoind: No such file or directory
When I run: which bitcoind
I get: /usr/local/bin/bitcoind

I have also tried to run: sudo ./bitcoind , sudo ./bitcoin-qt ,etc. with no other results. I have tried this in a tmp directory and a normal one.  

Started over with the proper ARM Linux download. Extracted the download and ran: sudo install -m 0755 -o root -g root -t /usr/local/bin bitcoin-0.20.1-arm-linux-gnueabihf.tar-3

ls -al /usr/local/bin returns "root root" for permissions on the bitcoin file in the directory

Would love some instructions specific to ARM Linux but none seem to exist.

I did try this and it returned "cannot execute binary file" This made me realize I had not installed ARM Linux version of Bitcoin Core. I have restarted a half dozen times with the proper download now and am encountering serious issues.

The primary concern is that I can't verify the download. sha256sum of the file I've downloaded does not match any on the SHA256SUMS.asc file I obtained from bitcoincore.org

The secondary concern is that the Linux install instructions at bitcoin.org/en/full0node#linux-instructions do not seem to apply to the ARM Linux version.

I've done a clean install of Ubuntu Desktop for Raspberry Pi4 with ARM architecture. I verified the download and am up and running just fine.
source: https://ubuntu.com/raspberry-pi

Now I try to download ARM Linux image of Bitcoin Core. (Trying to get bitcoind/bitcoin-cli up and running for a lightning node)
Verified this download and followed the instructions here: https://bitcoin.org/en/full-node#linux-instructions

Have no trouble extracting the file. Have no trouble running the command to move the file contents to /usr/local/bin (although it does not ask for my password)
Run cd ~/usr/local/bin/ to move into the directory
ls command shows the contents include bitcoind, bitcoin-cli, etc. (all contents of bitcoin-0.20.1/bin/ are in the proper /usr/local/bin directory)

However, when I try to run the bitcoind -daemon command I get an error along the lines of.

/usr/local/bin/bitcoind does not exist even though I can see it when I can navigate to the directory. I have exhausted all possible solutions by google search and have tried alternative install methods (Snap store install doesn't include bitcoind or bitcoin-cli)

I'm hoping this is a simple mistake