Well, if you saved during a Time Trial session, then it wouldn't be a time trial. But if you're saying it doesn't save your best times in Time Trial mode, that's sad. Unless the time trial is to see how fast you can get through the entire game.
You should be able to save in SMB3. Super Mario All-Stars let you save after every world. SMB3's not easy.
I look back at Mario Kart 64's time trial mode and wonder a few things. One, why did it take up almost all of my controller pak just to save a replay? Two, why couldn't you save if you crashed into something? Did it stop you from saving if you paused the game? The games I've seen since then make me wonder whether it was inefficient at saving replays or whether kart-driving and its tracks inherently take up more memory than, say, World Driver Championship.