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Here is another way to think about it: the perfect and the modal (including whether they are present or not) are parts of a complex grammatical system called tense–aspect–mood. In contrast, the subordinator to relates to a different aspect of verb grammar, that of finiteness. Thus postulating some construction of a form 'to/modal + have + past participle' mixes apples and oranges, i.e. conflates two very different issues.

Here is another way to think about it: the perfect and the modal are parts of a complex grammatical system called tense–aspect–mood. In contrast, the subordinator to relates to a different aspect of verb grammar, that of finiteness. Thus postulating some construction of a form 'to/modal + have + past participle' mixes apples and oranges, i.e. conflates two very different issues.

If you say that 'to/modal + have + past participle' is a single construction, you need to explain what makes it different from 'have + past participle'. In other words, you need to explain why the constructions 1. 'to + have + past participle' and 2. 'modal + have + past participle' are more closely linked with each other than either is with 3. 'nothing + have + past participle'.

Yes, you can say, "well, 'to/modal + have + past participle' has an extra word as compared to 'nothing + have + past participle'." But what matters for grammar is not the word count, but rather how a thing behaves grammatically, that is, what kinds grammatical functions it can fulfill and how it interacts with other grammatical objects. And in those respects, as should become obvious from the discussion below, there is nothing that links 'to + have + past participle' and 'modal + have + past participle' more closely than either is linked with 'nothing + have + past participle'.

In fact, if anything, it is 'to + have + past participle' that is the 'odd man out' here. As far as grammatical functions, 'modal + have + past participle' and 'nothing + have + past participle' are far more similar to each other than either is to 'to + have + past participle'. Namely, they both involve a finite verb, whereas the 'to' construction involves a non-finite verb. As we shall see, this is a very important difference as far as grammatical behavior.

If you say that 'to/modal + have + past participle' is a single construction, you need to explain what makes it different from 'have + past participle'. In other words, you need to explain why the constructions

[1] to + have + past participle

and

[2] modal + have + past participle

are more closely linked with each other than either is with

[3] (nothing +) have + past participle.

Yes, you can say, 'well, [1] and [2] have an extra word as compared with [3]'. But what matters for grammar is not the word count, but rather how a thing behaves grammatically, that is, what kinds grammatical functions it can fulfill—how it interacts with other grammatical objects. And in those respects, as should become obvious from the discussion below, there is nothing that links [1] and [2] more closely than either is linked with [3].

In fact, if anything, it is [1] that is the 'odd man out' here. As far as grammatical functions, [2] and [3] are far more similar to each other than either is to [1]. Namely, they both involve a finite verb, whereas the [1] involves a non-finite verb. As we shall see, this is a very important difference as far as grammatical behavior.

At this point you say, OK, so how about I just introduce the following more general construction: 'to/modal/nothing + have + past participle'. But this construction already has a name: it is simply what we call the (present) perfect tense, except in a form that makes explicit the well-known fact that the perfect tense can be used to mark both finite and non-finite clauses (see below) as well as clauses that may be marked for some kind of mood.

In fact, if anything, it is 'to + have + past participle' that is the 'odd man out' here. As far as grammatical functions, 'modal + have + past participle' and 'nothing + have + past participle' are far more similar to each other than either is to 'to + have + past participle'. Namely, they both involve a finite verb, whereas the 'to' construction involves a non-finite verb. As we shall see, this is a very important difference as far as grammatical behavior.

At this point you say, OK, so how about I just introduce the following more general construction: 'to/modal/nothing + have + past participle'. But this construction already has a name: it is simply what we call the (present) perfect tense, except in a form that makes explicit the well-known fact that the perfect tense can be used to mark both finite and non-finite clauses (see below) as well as clauses that may be marked for some kind of mood.