Since I'm not too fond of suppressing fields with en empty field format, here is a solution that redefines the involved commands directly.
\documentclass{article}
\usepackage[
citestyle=verbose-ibid,
bibstyle=authoryear-ibid,
]{biblatex}
\renewbibmacro*{date+extradate}{%
\iffieldundef{labelyear}
{}
{\printtext[parens]{\printdate}}}
\renewbibmacro*{bbx:ifmergeddate}[2]{#1}%
\addbibresource{biblatex-examples.bib}
\begin{document}
\autocite{knuth:ct:b,knuth:ct:c}
\printbibliography
\end{document}

A slightly less aggressive approach would still leave traces of labeldate by only redefining date+extradate as follows (and leaving bbx:ifmergeddate unchanged)
\renewbibmacro*{date+extradate}{%
\iffieldundef{labelyear}
{}
{\printtext[parens]{%
\iflabeldateisdate
{\printdate}
{\printlabeldate}}}}%
A different, but more direct, way to delete the extra date would be
\DeclareFieldInputHandler{extradate}{\def\NewValue{}}