I can't seem to find any way to put the cursor into 'overwrite' mode - as in when you press the insert key and newly typed characters overwrite the existing characters inline. I haven't found any reference anywhere online to the omission or inclusion of such a feature in VS Code, but it seems to be a fairly commonly used feature. Does this exist?
With version 1.96, VS Code now has native support for overtype mode.
No -- please vote for feature request Insert key doesn't switch to overtype/overwrite mode.
Meanwhile, you can try extension Overtype, last updated April 2023.