In practice it comes down to a few concrete choices rather than a tone of voice.
Timing. Most people working this channel mail off filed cases, and a case usually gets opened weeks or months after the death, often after the family has already decided someone needs to deal with the property. The filing itself is a signal that a person has taken on the job of resolving the estate.
Content. A letter that says who you are, that you buy houses in the area, that you can close without repairs or cleanout, and that you're happy to answer questions costs nothing emotionally. Deadline language and any suggestion that a decision has to be made now is what makes people angry, and it's also pointless, because nothing can close until the personal representative has letters and the state's required notice periods have run.
Taking no as final. Keep a suppression list and honor it. One letter to a house that never gets another one is a very different thing than eight.
The part that pays over years is that a lot of these calls aren't sales calls. Heirs ask what probate even is, who cleans out a house, whether they should just list it. Answering honestly, including telling someone their house is clean enough that a retail listing will net them more than you will, is what makes the attorney who referred them keep referring. Probate attorneys watch how their clients get treated.
If pressure worked here it still wouldn't work, because the court's calendar controls the closing date.
