Final minute clock stoppage question

[rulesmaven]

It is my understanding that a made basket that completes a play stops the clock if there is less than 1:00 in the game. (Or perhaps it is one minute or less.) But otherwise the clock runs after a made basket, and can continue to do so for several seconds (the time it takes to get the ball back to the inbounder subject to the official's discretion to stop the clock, plus up to five seconds for the inbound).

Do I have that right? It seemed to have led to an anomolous result in a game in one of the conference tournaments this year. A player scored with 1:01 left on the clock, and it was not until about 50.5 that the 35 second clock began on the possession when the player touched the inbound pass. Yet, if the first player had scored LATER, by a second, the clock would have stopped. That seems anomolous -- by scoring earlier, you run more time off the game. (The same thing can happen in football with the out-of-bounds clock stoppage under 5:00 minutes; if you're trying to conserve clock, you're better off going out of bounds with 4:59 than 5:01.)

Should there be a fix -- like an automatic stoppage at 1:00 until the ball is inbounded. Could this be feasibly done?