I have already made a comment about the definition of “simple sentence in this context.
The Oxford Grammar uses “simple sentence” to contrast with “compound sentence”. At 5.19 Greenbaum starts by distinguishing finite from non-finite verb phrases.
A verb is finite, if it displays tense, that is, the distinction between present and past.
He goes on to explain
A finite verb phrase can function as the verb phrase of a simple sentence, the verb of a main clause within a compound sentence, or the verb of a subordinate clause.
His example of a simple sentence makes it clear that he is using that terminology to indicate the absence of subordination.
Tonight I’m going to my first cocktail party at the Commission, my dearest.
His examples of compound sentences all involve subordinate clauses. But he goes on to say
A non-finite verb phrase cannot function as the verb of a simple sentence or as the verb of a main clause within a compound sentence. It can, however, function as the verb of a non-finite subordinate clause.
“[19] I don’t recall actually giving the name.
He goes on to explain
In [19] giving is the verb of an -ing participle clause. It is a transitive verb, and its direct object is the name.
It is interesting that he calls a participle clausewhat I have always thought of as a participle phrase. But I have argued in a comment that in OP’s question the participles are in effect functioning as verbs (which Greenbaum calls non-finite).
So I stick to my guns and say that the sentence quoted in the question is a compound sentence with subordinate clauses of which the verbs are non-finite.
