Is this somehow due to modal auxiliary verb, müssen?
Yes, it is.
Müssen governs the infinitive. The infinitive can be active or passive. In this case it's passive.
An example with the infinitive present active schneiden:
Ich muss meine Haare schneiden.
The infinitive present passive is:
to be cut
Meine Haare müssen ...
As RHa wrote, the sentence is grammatically correct. The first part results from reordering the words "Mehrere prominente Politiker haben ihre Teilnahme zugesagt". This would represent the subject–verb–object word order, which is the default order in the German main clause. However, German syntax can be flexible, especially, but not exclusively, ...
The sentence is grammatically correct. The explanation is that the perfect participle and the accusative object ("Ihre Teilnahme zugesagt") constitute a single sentence part. This allows them to occupy the first part of the sentence (the Vorfeld), right before the finite verb ("haben").
You can find more examples in the German Wikipedia ...
The development of the German language has been characterised by a weakening of unstressed syllables (Nebensilbenabschwächung). Looking at the 3rd person singular present indicative, Old High German suohhit – mahhot – saget became suochet – machet – saget, with reduction of the vowel to Schwa. Later, the vowel was syncopated, yielding New High German ...
This is in fact a sort of Zeugma (conglomeration of words that won't fit well), like in
Er schlug die Scheibe und den Weg nach Hause ein.
Ich heiße Heinz Erhardt und sie herzlich willkommen.
"Große und Hauptstadt Berlin" or "bayrische und Bierstadt München" would be of the same form, but barely acceptable by native speakers. The only ...