What grammatical roles do infinitives and participles assume when used predicatively?

Solution 1:

The question wouldn't arise if there weren't a false dichotomy between some uses of auxiliary verbs and others. Be is not a "linking verb". That's a term for grade school students. Be is an auxiliary verb. It is required whenever the main predicate is not a tensed verb:

• This gentleman is the one who phoned you. (predicate NP)
• That gentleman is rather ill. (predicate AdjP)
• The other gentleman is on the sofa. (predicate PP)

Of course, be is not the only auxiliary verb in English, but it is the most prominent one, and it's always an auxiliary verb. That is, it has no meaning; it's strictly part of the grammatical machinery, like the do in Do you like cilantro?.

Other uses of auxiliary be occur with various verb forms, participial and infinitive, in what are called Constructions in syntax. These are often called "tenses" or "voices" by grammar-school texts, but English only has two inflectional tenses (present and past), and no inflectional cases, moods, voices, or aspects at all, unlike other European languages. English does everything with constructions.

There is the Progressive (or Continuous) construction, which uses auxiliary be (in present or past tense), followed immediately by the present participle (the -ing form) of the next verb (which could be the main verb):

• He was looking at his phone.
• It's raining.
• I'm thinking about it.

Then there's the Passive construction, which uses be, also available in present or past, followed by the past participle (the -en/ed form, like taken or written, and often the same form as the past tense) of the next verb:

• His phone was stolen yesterday.
• This book was written by my friend.
• The children are fed by the nanny.

But those are in the verb chain and can be used together, along with modals and the Perfect construction, e.g,

• She will have been being photographed for 6 hours by then.

Auxiliary be doesn't occur with an infinitive in the verb chain; but it does occur in a specialized future construction for scheduled events:

• Frank is to leave tomorrow.

There's no special name for this construction, afaik; it's one of several ways of indicating the future in English.