I might be corrupted by programming because it doesn't bother me when I hear a vacuously true claim like "All flying horses are three-legged". My mental model corresponds to Python's definition of "all":
def all(xs, predicate):
for x in xs:
if not predicate(x):
return False
return True
Similarly,
def any(xs, predicate):
for x in xs:
if predicate(x):
return True
return False
You say "Ex falso quodlibet" which makes it sound like you believe such a system leads to contradictions, but this would come down to the choice of rewrite rules.
Example: a rewrite rule from "NOT all(xs, predicate)" to "any(xs, NOT predicate)" would lead to contradictions.
Reasoning:
A. all({}, predicate) is true by def of all
B. any({}, NOT predicate) is false by def of any
C. NOT all({}, predicate} is false follows from A
D. applying the rewrite rule to C, any({}, NOT predicate) is true, which contradicts B.
However, in first-order-logic, that same rewrite rule is fine since sets always contain elements. It doesn't seem like to much work to go from Python-esque any and all to ∀ and ∃. For example "all(xs, predicate)" could be rewritten to: "¬P ∨ (∀ xs. predicate)" where P is a new proposition variable for whether xs contains elements. This relies on the semantics including short-circuiting evaluation (which may be cheating, I don't know).
Or maybe the above is what you already meant when you said "following the embeding..."?