they were held off for 13 days. do you think santa ana wanted to sit around? was he on vacation? no. he had to wait until he felt he had enough of an advantage to attack. it took 13 days for him to feel confident enough. if he thought he could he would have tried to take it the first day.
Santa Ana would only have made an assault on the first day if there was an overwhelming strategic need for a quick victory.
Why would he try to take it on the first day against fresh defenders ? He would have been stupid to try, it would have caused uneccessary casualities in his own army. Much cheaper and easier to take it after the defenders were demorolised by lack of supplies, food and powder.
There is no doubt that the "Hollywood Alamo" bore little resemblance to the real battle and the "fight to the last man" is propaganda.