First and foremost, thank you for choosing justanswer.com. I am a licensed criminal defense attorney and member of the American Bar Association.
The "speedy trial" in Texas is quite a lucid concept. The speedy trial is defined by the Texas courts as follows under the four-factor balancing test set out in Barker v. Wingo, 407 U.S. 514 (1972), which requires balancing the following factors: (1) length of delay, (2) reason for delay, (3) assertion of the right, and (4) prejudice.
Obviously, your brother in law would have an argument that his "speedy trial" rights were abused due to his attorney's seemingly incompetent performance. Of course, this would depend upon whether or not there is actual hard proof his attorney has "forgotten" court dates or has simply been too "drunk" to show up.
The above could be hard to prove to a Texas Appellate court. Also, you have to consider the time it would take to follow through the trial process with the current attorney, and actually get to the appellate process.
The best move would be for your brother in law to have a motion filed for change of counsel. The most important evidence would be the change of court dates without any particular reason by his attorney. Your brother in law's attorney would have to answer for the reasons the continuances were filed and, assuming your accusations are correct, there should be be enough reason for a change of counsel to be granted.
You ask if requesting a new trial would do away with the speedy trial right: the answer is yes and no. Requesting new counsel does not due away with the speedy trial right; that right is absolute. However, any continuance granted for your brother in law to obtain new counsel, private or court appointed, would run against him. Still . . . the benefit of a competent attorney will outweigh the time spent in obtaining one.
I hope this helps. Please, click "Accept" if you found my answer helpful. Do not hesitate to ask follow up questions after accepting my answer as I will be more than happy to offer further help. Thank you in advance and good luck to you and your brother in law!