The cheap talk problem
Economists have a term for statements that cost nothing to make: cheap talk. Because anyone can say anything for free, a listener has no way to separate the sincere from the strategic. The rational response is to discount everything.
That is exactly what happens when someone reads the fifth apology text from a number they were about to block. The words might be perfect. The problem is the words are free.
Costly signalling in one paragraph
A signal becomes believable when it is expensive for a liar and affordable for someone honest. Engagement rings, long commutes to see a friend in hospital, and quitting a job to help family are all read as proof precisely because they hurt. Biologists call it the handicap principle. Everyone else calls it actions speaking louder than words.
What the recipient actually learns from an amount
When someone sees that a message was paid for, they learn three things before reading a single word. The sender thought about this for longer than a text takes. The sender accepted a loss with no guaranteed return. The sender expects the message to be refused and sent it anyway.
None of those facts guarantee the message is worth reading. All of them raise the odds enough for a person to give it thirty seconds.
The line between a signal and a bribe
A bribe pays the other person to act against their own judgement. A signal costs the sender and pays the recipient nothing. That difference is the whole design question, and it is why the money in a paid message request should never be a payment to the recipient in exchange for reading.
The right frame is simple. You paid to be allowed to ask. They still owe you nothing.
How much is the right amount
The amount should be uncomfortable for you and legible to them. Too small and it reads as a gesture. Too large and it reads as pressure, which can make a person feel bought rather than asked. Most people land somewhere that genuinely stings for a week and no longer.