The ablative absolute does not require a participle. It can be a noun and an adjective, as you say, or two nouns (Caesare duce urbem cepimus), or even an adjective and an accusative with infinitive (most probably, see the end of this post).
However, there is a "verb-like" aspect to the construction that makes you want to add "being" if there is no participle. The reason for this probably lies in the fact that the ablative absolute is a so-called dominant construction, where semantics and syntax are slightly out of tune: semantically, there is a kind of 'oblique predicate' (with a 'subject' and its 'predicate'), even though syntactically there are just two nominals in the ablative. And for there to be a predicate, you need a kind of verb in your translation.
It is called dominant because the participle in a dominant construction is normally translated more prominently than a mere (attributive) participle: it is translated as a finite verb (or a noun). Otherwise our translation would fail:
Legione dispersa victi sumus.
If you translated this as "with the dispersed legion, we were defeated", that would sound wrong, because the participle is translated attributively. English has a trick where you can make an adjective or participle predicate-like by placing it after its 'subject': "with the legion dispersed, we were defeated". That works, because the participle is translated predicatively. The standard translation would be "after the legion was dispersed, we were defeated". This "the legion was dispersed" is a full predicate.
Caesare duce urbem cepimus.
"With our leader Caesar, we captured the city". That is grammatical, but it is not what the Latin intended, if only because with is usually rendered as cum when followed by a person. It should be "with Caesar as our leader, we captured the city". The word as (rightfully) introduces a predicate-like aspect to our translation: one of the things we want to say is that Caesar was our leader. That is a fact that the Latin is stating separately by means of the ablative absolute, supporting the 'main clause' urbem cepimus. You could also translate it as "because Caesar was our leader, we captured the city" (optional causal link), with a full predicate.
Other examples of dominant and predicative constructions are the gerundive construction, the so-called dominant participle (of which you could say the ablative absolute with a participle is a sub-species), the dominant substantive apposition (of which the ablative absolute without a participle is a sub-species), and even the ordinary copula with subject complement.
Floribus carpendis Eurydice mordetur a serpente.
"During/with the flowers to pick, Eurydice is bitten by a serpent." That's not right. The gerundive construction should be translated dominantly, where the adjectival/attributive gerundive is translated as a substantive noun or even as a finite verb: "during the picking of flowers, Eurydice is bitten by a serpent", or "when she was picking flowers, Eurydice was bitten by a serpent".
A Latin alternative is the gerund construction, which is normal, not 'dominant', because a gerund is a kind of noun, which is already fairly dominant: as such, the gerund is not translated as more dominant than a noun normally is.
Flores carpendo Eurydice mordetur a serpente.
"During the flowers-picking, Eurydice is bitten by a serpent."
Caesar occisus senatoribus placuit.
"The dead Caesar pleased the senators." That kind of works, because a dominant participle construction also happens to be possible in English in certain cases. But what we're really saying is this: "Caesar's death pleased the senators". A substantive noun is stronger than a mere attributive participle. Or, even more dominantly: "the fact that Caesar died pleased the senators". What we're really not trying to say is "Caesar (who had died) pleased the senators". The brackets express a low degree of dominance, since the words are thereby separated from the main sentence.
Caesar primus Romam advenit.
"The first Caesar arrived in Rome": it seems clear that this is not what was intended. "Caesar arrived in Rome first": this predicative translation is correct, where the predicative adjective is translated almost like a subject complement or an adverb of manner.
Caesar victor Romam advenit.
"The victor Caesar arrived in Rome." This is perhaps acceptable, but what the author probably intended was the predicative translation "Caesar arrived in Rome as (the) victor". The word as clarifies that it is to be understood predicatively.
Caesar fortis est.
This is a plain copula with a subject complement. An attributive translation would be absurd: "the brave Caesar is". Copula with subject complement is the simplest kind of predicative construction, equally common in other Indo-European languages. We must translate the sentence predicatively, i.e. as a copula with subject complement: "Caesar is brave".
N.B. There is one kind of ablative absolute that interests me:
Audito Agrippinam a filio suo necatam esse tota in urbe populus luxerunt.
"When it had been heard that Agrippina had been murdered by her son, the people mourned throughout the city." The participle in the ablative is the neuter audito; the 'noun' belonging to the ablative absolute is the entire accusative with infinitive, Agrippinam a filio suo necatam esse.
This is possible in Latin, and I believe it is used when you need an accusative with infinitive but the verb of saying or hearing or thinking etc. cannot be made to agree with an existing word in the sentence. This may happen when the hearing happened before the main clause, because then you need a past participle, and past participles cannot be made to agree with an agent (because they're passive). The person who hears cannot be made to agree in number, case and gender with the past participle auditus, so it has to be used absolutely.
A quotation from Quintus Curtius Rufus:
Alexander audito Dareum movisse ab Ecbatanis, omisso itinere, quod petebat, in Mediam, fugientem insequi pergit strenue.
— Quintus Curtius Rufus, Historiae Alexandri Magni 188.8.131.52
"Alexander, when it had been heard that Darius had moved away from Ecbatana, ..." (more liberally translated as "when he had heard that...").