刘老师大意了。助动词包括基本助动词（do ,have, be)和情态助动词. 基本助动词中的be和have都可以有非谓语形式的。

Having finished his homework，Tom went out to play. Having是助动词have的现在分词。

She hates being laughted at by others. 动名词被动语态作宾语。being是助动词be的动名词形式。

I‘m so glad to have shaked hands with President Xi. to have为助动词have的不定式形式。

网友句子中的being是系动词而不是助动词be的动名词，是因为welcoming前的程度副词so明确表示welcoming是形容词作表语。所以being是系词，而不是助动词。