In the 1928 novel The Well of Loneliness, Radclyffe Hall writes:
Arrived at the church, she and Wanda would stand looking down between the tall, massive columns of the porch, on a Paris of domes and mists, only half revealed by the fitful sunshine.
The arrived at the church struck me as equivalent to formulations such as [having] arrived at the church or [after they] arrived at the church, and when I first read it, it sounded awkward.
With some experimentation, I noticed that there were some cases where an auxiliary have could be omitted in a modifiers of this variety. I am not considering cases where the modifier clearly has a passive voice construction and where be is omitted. (An example of a construction that clearly uses passive voice: [Since she was] worked to the bone, she quickly ended up in a psychiatric ward. Since she was can be omitted without issue.) Examples include:
[After she/Having] Graduated from the university, she was left without a safety net.
[Having] Departed for work, he did not see his house burn to ashes.
Meanwhile, certain constructions are impermissible and require have:
Eaten the whole cake, he was stuffed.
Slept like a baby, she was well-rested the following day.
Sped while driving through the neighborhood, he got a ticket.
Constructions with been or become — e.g. Been sick for the last ten days, she was very behind on work. — are also unacceptable. (been appears to be special; one solution there is simply to drop been as well.) So are constructions with verbs related to wanting something:
Wanted to study at Chicago for a long time, he was disappointed when he learned that his application had been rejected.
Alternatively: Wished for her son to study at Chicago for a long time, she was disappointed …
Hoped that direct elections would eventually come, she was angry about the Central Government's decision.
What exactly governs the acceptability of these constructions, where auxiliary have is omitted before a past participle in a modifying phrase? (For instance, does this have anything to do with certain grammatical qualities of the verb being used? Alternatively, is my analysis of the acceptable constructions missing something?) I'm having trouble pinpointing that, so I'd like some input.