That is a reasonable way of thinking about it.
With the fall of the Roman Empire, cities were abandoned as centers of administration.
My grandfather fought in World War II as an infantryman.
With verbs, we can distinguish between verbs which take Objects and verbs which take Predicative Complements:
- The doorman ejected the politician. (the politician = object)
- The doorman was a politician. (the politician = predicative complement)
In (3) there are two people involved, the doorman and the politician. In this particular example, the doorman happens to be doing something to the politician, but this is incidental. Direct Objects are Complements of the verb which introduce a new entity into the clause (my own definition), but they do not have to have any particular semantic relationship with the Subject, and the Subject does not need to be doing anything to them. This last bit is not my definition but a widely known fact; consider 'She survived her husband by twenty-five years.'
Example (4) is quite different, and only talks about one individual. Here, the noun phrase a politician describes the Subject, the doorman. As a Predicative Complement, a noun phrase does not introduce a new entity into the clause, but rather, describes an existing one (or existing ones). It may describe an attribute of the Subject, or explain who or what the Subject was, or describe a role that they had.
Predicative Complements often relate to Subjects as in (4), but they can also relate to Objects as in (5-6):
- They elected Maria President.
- The exam process made Bob a nervous wreck.
In (5) President describes the role of Maria, the (denotee of the) Object of the verb elect. In (6), similarly, the term a nervous wreck applies to Bob (not the exam process).
Importantly for us here, a very few prepositions can also take noun phrases (as well as adjectives) as Predicative Complements.
- She served as the president.
- She was regarded as the favourite.
- I had a fool for an advisor.
In his (2023) paper Oblique predicative constructions in English with for and as : qua vs qualitate qua, Bas Aarts calls such constructions ᴏʙʟɪᴏ̨ᴜᴇ ᴘʀᴇᴅɪᴄᴀᴛɪᴠᴇ ᴄᴏɴsᴛʀᴜᴄᴛɪᴏɴs. That paper has considerable discussion of the difference in meaning between oblique predicatives with for and as. The meaning given for for-predicatives is very difficult to describe in a clear and succinct way1. However, Aarts notes that predicatives of the form as X, normally mean in the role or capacity of X. This chimes with a useful comment under the question by @TimR, which I hereby preserve for posterity:
He used the rag as a mop. I don't think as a mop describes the rag but qualifies the nature of the use. In both cases in the question, the PP refers to a role, which is not an inherent attribute. The role of the cities had been centers of administration. The role of granddad was infantryman. It is how the cities and granddad were put to use.
I'm not sure this applies to all uses of as-predicatives, but it is apt for the Original Poster's examples.
Does the use of an oblique predicative turn the verb into a linking verb? I'd say not. However, Aarts notes that
Grammars of English generally regard the oblique predicative construction as more or
less identical to the ‘ordinary’ predicative constructions shown in (1)–(4), or at least as
‘parallel’ (Jespersen 1909–49, IV: 375).
This is not quite the same thing. It says that the constructions are similar, not that the verbs are both linking verbs. It's probably better to think of as as a linking preposition. In any case, I'm not sure if the constructions really should be thought of as parallel in any way other than semantically. But intelligent and reasonable people can disagree about this.
1Not least because the only paraphrase given in simple English uses the preposition as!