The Avudraham's opinion that one would be required to fast on 10 Teves even if it fell on Shabbos is well known (see here and here). The same is not true even of 9 Av. Why is 10 Teves more significant than other fast days? Why should the beginning of the siege process that years later led to churban be more significant than the churban itself?
Chasam Sofer explains that 17 Tamuz is a fast which commemorates past events – the walls of Yerushalayim were breached. 9 Av is a fast which commemorates past events – the churban, among other tragedies, took place. Same for the fast of Gedlaya. Not so the fast of 10 Teves. True, the siege was put in place on 10 Teves, but other enemies has also laid siege to Yerushalayim and they were defeated. There was time yet to avert a churban. The fast of 10 Teves is not a fast that commemorates events which already occurred, but is rather a fast of an eis tzarah, a fast to avert future tragedy.
The failure to rebuild the Mikdash is tantamount to witnessing its destruction. The din v’cheshbon of whether this year will be another year of continued churban or whether this year will be the year we avert 9 Av and witness the rebuilding of the Mikdash occurs on 10 Teves. The future is in our hands to determine.