Both are fine, are grammatical, and are standard English usage.
In your two examples, the expressions "a broken man" are predicative and are functioning as predicative complements (PC).
Here are some related examples:
- CGEL, page 261, : He died young.
The "young" in that example has a depictive interpretation.
Examples that show the "as" phrase as having a PC as its complement:
CGEL, page 636, [4.i]: I regard their behaviour [as outrageous].
2005 textbook, page 140, [27.ii]: I regarded her [as a friend].
There are a lot of verbs that take depictive PCs. CGEL, page 263, :
Kim felt [lonely / an intruder].
Her son remained [ill / a danger].
That seems [plausible / a good idea].
Pat proved [reliable / a great asset].
And some more verbs, taken from CGEL, page 263, : feel, look, smell, sound, taste, continue, keep, remain, stay, appear, seem, prove.
Those examples are from CGEL "5.4 Classification of verbs taking predicative complements", subsection "Class 1 verbs: complex-intransitives with depictive PCs". And in "5.4" are more subsections with more different types of classes of verbs taking PCs, on pages 263-266.
(Note: "CGEL" is the 2002 reference grammar by Huddleston and Pullum et al., The Cambridge Grammar of the English Language. "2005 textbook" is the 2005 textbook by Huddleston and Pullum, A Student's Introduction to English Grammar.)
EDITED: In light of a recent comment, here's some additional info as to the use of the preposition "as" in taking PCs -- CGEL page 636,
(b) Predicative complements
The main preposition taking a predicative complement is as: this is the prepositional analogue of the verb be. The whole as phrase may function as complement or adjunct in the larger construction containing it:
The complement is predicative in that it is related to a predicand: the object their behaviour in [i], the subject I in [ii]. In the complement use, the as is selected by the verb -- in this example, by the prepositional verb regard (see Ch. 4, &6.1.2).
If you're interested in using Google Ngram Viewer, then perhaps compare the string "died a * man" vs "died as a * man". In that graph, notice the similar string "died as a young man" and its usage in the year 2000.
To my AmE ear, the versions using "as" are just as good as the versions that don't (caveat: when the "as" is selected by the verb). (Though, for a literary type of register, I can see why authors and editors might more often prefer the versions that don't use the preposition "as".)