A simple option to this is to set up a Macro in a program like
Keyboard Maestro or create a script in
Pastebot such that some key AND clicking the status (menbar) itemwill trigger "insert text by typing". You will note that a lot of programs have problems with various key combinations in Linux or Windows, so it is best to use an F key that is outside of normal definition and the menubar item as a backup:
NB: the delay in there helps to reduce problems in some scenarios. For example, when I use this to "paste" text into a Proxmox virtual machine and have it run at full speed sans delay, it will somehow corrupt the text that is inserted sometimes. With this delay it seems to reliably enter the correct clipboard/pasteboard contents.