Seller accepted our financed offer over alleged all-cash offer: is it a red flag?

We offered on a property listed at ~$1.27M. Our initial offer of $1.275M was countered, seller said they needed more. We came up to $1.28M and were accepted over a competing cash buyer (allegedly higher) because we could close faster.

A few days into due diligence, the seller’s agent told our agent that another all-cash offer had come in $30K above ours at $1.31M and could match our closing timeline. Rather than walk, we went to $1.32M with an appraisal floor contingency well below our offer price. Accepted the next day.

Questions: (1) How common is it for a financed offer with an appraisal contingency to beat an all-cash offer that’s only $10K less? At this price point that spread seems too small for cash to lose. (2) Is the timing suspicious:a new cash buyer appearing right after we started DD? (3) Should we have asked the seller’s agent for proof of the competing offer? (4) Are we overthinking this, or is this a standard bidding-up tactic?

Thanks in advance for your honest opinion.