No you would not have to sit there doing nothing. You can sing hymns, join in prayer, etc. (Assuming the songs and prayers are not in opposition to the Catholic faith… I’m thinking of fringe groups like LDS, Jehovah Witnesses, and non-Trinitarian churches).
You should refrain from doing anything that clearly denies or goes against a Catholic teaching or prohibition in canon law-- for example you cannot participate in non-Catholic communion, a “same sex” marriage, being a “godparent” to a non-Catholic child, etc.
This Catholic Answers Q&A you link to is NOT an authoritative teaching of the Church. It is a Catholic Answers staff member opinion.
There is a definite difference between a one time special event like a family/friend wedding, baptism, funeral, and regular attendance at a non-Catholic service. I think the Catholic Answers response is a little off. Catholics are not prohibited from attending non-Catholic services on an exceptional basis for reasons such as the one you mention. Nor is there any prohibition on praying, singing, etc, while attending.
For Church teaching, look to the Vatican II documents on ecumenism and also to documents such as Ecclesia de Eucharistia, Ut Unum Sint, Dominus Iesus, all available at the Vatican website.