Are there many sites that only support Authy's push authentication and nothing else?

No, not really, it's about migrating/restoring every single 2FA key would be extremely inconvenient

TOTP should just be a (typically base32) secret string; I don't know if Authy allows exporting that though (and if not, that only underscores the point of this article).

I just use a simple shell script with dmenu/xclip/oathtool:

  #!/bin/zsh

  typeset -A opt=(
      Docker ABC
      GitHub DEF
      # ...
  )
  k=$(print -l ${(ko)opt} | dmenu -i)
  [[ $k != "" ]] && oathtool --totp --base32 $opt[$k] | xclip -rmlastnl

Do one a day :-)