But consider some further implications:

If the woman is the purchaser you must say,
The woman and its husband went shopping, and it bought a pair of new pants.
While if the man is the purchaser, you must say,
The man and its wife went shopping, and it bought a pair of new pants.

This creates at least two problems.
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First, you can't gracefully handle a more complex situation, where the second clause covers both people. Such as
The woman and her husband went shopping, and she bought him a new pair of pants. or
The woman and its husband went shopping, and he bought a shirt while she bought pants.
If you lack gender-specific pronouns, you must resort to clumsy substitution of "the former" and "the latter".
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Second and more important: even in the simpler sentences, your phrasing of the first clause commits you to the meaning of the second, even before the second is uttered. Imagine the difficulty in an ongoing conversation, where each idea leads to another. E.g., two women chatting:

"Marge and John went shopping."
"Where?"
"At the mall. There was a *delicious sale."
"Did they buy anything?"
"Yes, it bought a new pair of pants."

"Who did? I can't remember exactly how you phrased your first sentence before."
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Doesn't gender in pronouns make it far easier to minimize such confusion?