When we bought our house (spring 2019, so add two more years of hysteria on that), on top of the usual written offer, they also opened the door on the evening of receiving offers for the purchasing realtors to make a pitch on top of the pure monetary offer. Our realtor didn't think anyone else would show up, so thought it would help. Turns out so did seven other realtors, so it apparently was like an audition with all the realtors sitting in a waiting room, having their names called one by one. Of the seven who showed, only two left right away, suggesting a close race. We ended up getting it partly because we were happy with a very long close, which shows it can be more than just about absolute dollars.
Assuming offers are close, the letter may help on a number of fronts. As mentioned, highlighting financial solidity helps, as it's not unheard of for deals to fall through with buyers who can't get what they hope they can. This used to be an issue for sellers, as a relisted house used to be less appealing than a new listing for the usual odd psychological reasons. I suspect it doesn't much matter in today's low inventory feeding frenzy, though.
The other factor is if people have lived in an area for a while, they've likely formed friendships people close by and don't want to saddle them with bad neighbours or a rental. When we sold in BC, we definitely considered how the buyers would fit the area, as we got an immediate bully offer and had to decide whether to accept or hold out for more showings. Having met them during the rush showing (much to both realtors' chagrin) we thought we had a good sense of what they were about and it made accepting the offer easier.
As an aside, for the usual flood of flyers and calls from flippers and realtors desperate for listings, we got a letter in our mailbox last night from a woman and her family describing how they loved the neighbourhood, how they were desperate to live here, and offering all sorts of reasons why a private sale to them would be preferential to a typical listing (no fluffing, no realtor commission, no open houses, solid financials, long close, price agreed based on recent comparables, etc.) It's possible it was a clever flipper, but they definitely leaned heavily on how they would be great additions to the area and how they would care for the house...