The lower South T glacier is technically "closed," due to the C-47 WWII plane that crashed there years ago...although it's not inforced. I don't think there's ever been any issues with climbers accessing other routes across the glacier (i.e., South T headwall).
The route is steep, reasonable, with no resting places for 4k of elevation gain. Best conditions are probably in late April through June...combined with a cold snap and reasonably firm snow. Move quickly and beware of rock and ice fall. Either climb unroped...or simul...with running belays (probably pickets).
Some NPS rangers have climbed it in recent years.
South Tahoma Glacier and the Tahoma Glacier are different routes. Post your TR and photos if you do it. I'll look for a photo...