When I upgraded to Big Sur, I had the same issue. It's something to do with npm, not react-native.
Here's what I did to solve the problem:
Delete everything related to nodejs:
sudo rm -rf ~/.npm ~/.nvm ~/node_modules ~/.node-gyp ~/.npmrc ~/.node_repl_history
sudo rm -rf /usr/local/bin/npm /usr/local/bin/node-debug /usr/local/bin/node /usr/local/bin/node-gyp
sudo rm -rf /usr/local/share/man/man1/node* /usr/local/share/man/man1/npm*
sudo rm -rf /usr/local/include/node /usr/local/include/node_modules
sudo rm -rf /usr/local/lib/node /usr/local/lib/node_modules /usr/local/lib/dtrace/node.d
sudo rm -rf /opt/local/include/node /opt/local/bin/node /opt/local/lib/node
sudo rm -rf /usr/local/share/doc/node
sudo rm -rf /usr/local/share/systemtap/tapset/node.stp
brew uninstall node
brew doctor
brew cleanup --prune-prefix
And then install nvm again via:
curl -o- https://raw.githubusercontent.com/nvm-sh/nvm/v0.37.2/install.sh | bash
Now, I was able to install all of my dependencies since now I've got a fresh copy of npm(for me it was 6.4.1).