Obviously, in cases like Greg Lee's above, a non-finite clause introduced by the 'complementizers' with or without simply cannot be controlled, because 'Control', as defined in GB, P&P and subsequent Chomskian metatheory, does not apply to 'referential' NP/DPs like the onlookers, which, by definition, must be autonomous in reference (recall Principle C of Binding Theory), but, assuming you are interested only in non-finite predications containing PRO subjects, the answer is still No, non-finite adverbial clauses do not have to be 'controlled', because their PRO subject can - if only in exceptional cases - be arbitrary in reference, as in e.g. PRO Talking about serious matters now, what do you think of Trump's attitude towards Russia and Putin? Of course, in all the examples you provide in your question (and you could have added predications apparently consisting only of PPs, NPs, etc.), PRO subjects are controlled, and they are controlled by the subject of the 'main clause', but there is no choice: such PROs cannot be controlled by NP/DPs in any other position, because 'control' requires c-command and no NP/DP can c-command a detached adjunct from any lower position than that of subject. As to whether Green with envy, Bound in leather, etc. could be analysed other than as bare (or 'small', 'reduced', 'verbless' etc.) 'clauses' (i.e.,'predications'), I agree with you that, from a Chomskian perspective, they cannot, but, even if, according to some other grammatical metatheory, they could be considered detached 'predicative' APs (PPs, NPs, etc.), instead of 'clauses', the fact would remain that green, bound, etc. are 'open', one-place 'predicates', and, unless a 'predicate' has a 'subject' argument, it is, by definition, 'unsaturated' and uninterpretable (in the sense that any unvalued variable makes the value of the function undefinable). Hence, even if other grammatical metatheories are entitled to object to 'empty' categories like Chomsky's PRO, they still have to say somehow that the 'external' argument of green, bound, etc. (or its theta-role) is 'discharged' ('satisfied', 'saturated',... you name it) by the entity referred to by the subject of the main clause, which makes such alternative solutions and Chomsky's original one mere notational variants of each other.

Obviously, in cases like Greg Lee's above, a non-finite clause introduced by the 'complementizers' with or without simply cannot be controlled, because 'Control', as defined in GB, P&P and subsequent Chomskian metatheory, does not apply to 'referential' NP/DPs like the onlookers, which, by definition, must be autonomous in reference (recall Principle C of Binding Theory), but, assuming you are interested only in non-finite predications containing PRO subjects, the answer is still No, non-finite adverbial clauses do not have to be 'controlled', because their PRO subject can - if only in exceptional cases - be arbitrary in reference, as in e.g. PRO Talking about serious matters now, what do you think of Trump's attitude towards Russia and Putin? Of course, in all the examples you provide in your question (and you could have added predications apparently consisting only of PPs, NPs, etc.), PRO subjects are controlled, and they are controlled by the subject of the 'main clause', but there is no choice: such PROs cannot be controlled by NP/DPs in any other position, because 'control' requires c-command and no NP/DP can c-command a detached adjunct from any lower position than that of subject. As to whether Green with envy, Bound in leather, etc. could be analysed other than as bare (or 'small', 'reduced', 'verbless' etc.) 'clauses' (i.e.,'predications'), I agree with you that, from a Chomskian perspective, they cannot, but, even if, according to some other grammatical metatheory, they could be considered detached 'predicative' APs (PPs, NPs, etc.), instead of 'clauses', the fact would remain that green, bound, etc. are 'open', one-place 'predicates', and, unless a 'predicate' has a 'subject' argument, it is, by definition, 'unsaturated' and uninterpretable (in the sense that any unvalued variable makes the value of the function undefinable). Hence, even if other grammatical metatheories are entitled to object to 'empty' categories like Chomsky's PRO, they still have to say somehow that the 'external' argument of green, bound, etc. (or its theta-role) is 'discharged' ('satisfied', 'saturated',... you name it) by the entity referred to by the subject of the main clause, which makes such alternative solutions and Chomsky's original one mere notational variants of each other at best! (i.e., provided the alternative solutions represent co-reference, c-command restrictions, and other principles of Control Theory in terms comparable in rigor and accuracy to Chomsky's own).