I am not sure whether participle or gerund of modal verbs is possible in english. For example we have the sentence: Having introduced the speaker, the chairman opened the meeting. What would be the correct form when using the modal verb "must"? Is it "having had to introduce the speaker…"?
You have it exactly right. The full modals (can/could,may/might,must,shall/should,will/would) are 'defective': they have no non-finite forms (participles and infinitives) and do not inflect for person and number. That's one reason why the 'periphrastic modals' like HAVE to, BE going to, BE able to have increased in use over the last couple of centuries.