No, Muslims will never say, " For Muhammad's sake" because we don't consider Muhammad as a God but Muslims are encouraged to use Subhanallah, Masha Allah, Alhamdulillah, Allahuakbar, innalillah...(it's a bit longer), Lahaulawallah.... (also it's a bit longer), Astaghfirullah and Lailahaillallah or simply Ya Allah but they are not compulsory. These are not curse words. All of these words are known as dzikr/zikir.
We say Subhanallah( Glory be to God) and masha Allah when we see beautiful/(or otherwise) things created by The God like natures or to express happiness.
We say Alhamdulillah(Praise be to God) when we feel grateful or when we get what we want.
We say Allahuakbar(God is The Greatest) to give us strength.
We say Innalillah...(From God we come and to God we will go back) to express sadness or when we hear the death of someone.
We say Lahaulawallah...( There's no power without God's help) when we seek the help from God.
We say Astaghfirullah when we're asking a pardon from God.
We say Lailahailallah- meaning There is no God other that the true AlMighty. To remind us that there is no other god but the one true non-being God that is to be worshipped alone.
Ya Allah simply means Oh God. The simplest expression. I always use this every day. And 99% other Muslims that I have met will always use this even though they are non-practicing Muslims.
Most Muslims say this every day including me. Sadly some bad people use this beautiful meaning expressions when they do bad things to the point that people who don't know the true meaning of these words associate the words with badness.
We are all allowed to say these words because they are not curse words and Muslims will not be offended at all.
That Muslims gentleman could have said Astaghfirullah or simply Ya Allah in his situation.
Hope this explain as to why you never hear people say For Muhammad's sake.