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
 to + have + past participle
 modal + have + past participle
are more closely linked with each other than either is with
 (nothing +) have + past participle.
Yes, you can say, 'well,  and  have an extra word as compared with '. 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  and  more closely than either is linked with .
In fact, if anything, it is  that is the 'odd man out' here. As far as grammatical functions,  and  are far more similar to each other than either is to . Namely, they both involve a finite verb, whereas the  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.
There are good reasons to distinguish constructions involving the subordinator to from those involving auxiliaries (including modal auxiliaries).
The main issue is whether the verb is finite or non-finite. These two have vastly different grammatical properties, as I will explain below. But 'modal + have + past participle' is finite, whereas 'to + have + past participle' is non-finite. So it makes no sense to speak of a single construction 'to/modal + have + past participle'.
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.
From this perspective, all your examples show is that the marking for the perfect tense may occur in both infinitival (thus non-finite) clauses and in finite clauses that are also marked for some kind of mood.
Infinitives (or infinitivals in the terminology of CGEL), whether with the to or bare, are non-finite. They may be marked for perfect tense, progressive aspect, passive voice, or (in principle) any combination of those. That gives these eight possibilities (see also here):
(to) eat (plain infinitive without further markings)
(to) have eaten (perfect)
(to) be eating (progressive)
(to) be eaten (passive)
(to) have been eating (perfect progressive)
(to) have been eaten (perfect passive)
(to) be being eaten (progressive passive; for an example of usage, see here)
(to) have been being eaten (perfect progressive passive; quite possibly never used in actual speech or writing)
Already at this stage one should start suspecting that the construction 'to + have + past participle' is an infinitive first, and a perfect second. In other words, distingishing phrases according to whether they are perfect or not is secondary to distinguishing them according to whether they are infinitive or not. And if so, then we should not attempt to lump together constructions with to and constructions with a modal. However, let's continue with the discussion.
All eight of these are called non-finite clauses. Besides the infinitivals, there are two other form-types of non-finite clauses (CGEL, p. 1173). In the examples that follow they appear in brackets: gerund-participial (I remember [locking the door]), and past-participial (His father got [charged with manslaughter]).
All non-finites characteristically enter into catenative constructions; indeed, only non-finites can enter such constructions (CGEL, p. 1177). The term ‘catenative’ (from Latin catenatus, “chained”) is used because of the possibitilty of chains in such constructions, e.g. We need [to go to the tennis court] [to help Jim] [to get some practice].
Non-finite clauses can also fulfill various other grammatical functions (CGEL, pp. 1251-1266):
subject: [To refuse her request] would be unthinkable.
object: This made [obtaining a loan] virtually impossible.
Note that an infinitival can only appear as a so-called extraposed object: I thought it wise [to adopt a low profile].
predicative complement: His intention was [to begin at six].
complement of an adjective: You are free [to leave when you want].
complement of a noun: She had the strength [to withstand this pressure].
complement in the structure of a preposition phrase (PP): We got up at five [in order [to catch the early train] ].
indirect licencing (here by enough): Enough people turned up [to form a quorum].
matrix-licensed complements: You have no choice but [to accept her offer].
interrogative clauses: I dont know whether [to go].
post-head modifier in noun-phrase (NP) structure: This provides [a solid foundation [to build on] ].
modifier in clause structure: They are saving up [to buy a washing-machine].
supplement: [To put it bluntly], they're utterly incompetent.
There is in principle a vast amount to be said about finite verbs. But for our present purposes, the following observation will suffice: many of the functions we have just listed for non-finite verbs are simply not possible for finite verbs, regardless of whether the finite verb is marked for the perfect tense or not.
Take the example of should have received a letter. It cannot function as a catenative complement, nor as a subject, nor as an object, nor as a complement of an adjective, etc. In contrast, to have received a letter can appear in all those functions:
catenative: I happened [to have received a letter].
subject: [To have received a letter] would have been nice.
extraposed object: I thought it nice [to have received a letter].
complemet of an adjective: He is content [to have received a letter].
Thus it makes little grammatical sense to speak of a 'to/modal + have + past participle' construction: the construction with to is grammatically vastly different from the construction with a modal.