If the roleplaying elements of going through a trial are a known problem with some members of the group, but the trial itself is too important to gloss over the way the trial is presented will have to engage everyone from the start. If it begins the same way as previous - unappealing - encounters with the legal system, it will be harder to keep things moving. As your plan also involves three such encounters, this element takes on even more importance.
The idea of using Jenga has a lot of merit as it clearly signals a fairly short duration to the trial sequence, and brings a certain degree of tension all of its own. While some may find it distracting or a problem for immersive roleplay - in the situation described, roleplay in this type of situation/environment is already a problem for a part of the group, and so trying something different like this may balance the scales.
Phase One: Building a Case
The prosecution has to demonstrate a reasonable case against the accused. In this phase that side (the GM) has to build their argument while building a house of cards. Each successful layer or set of layers in this card house will represent the initial pool of "points" that the defense will have to overcome in order to maintain the accused's innocence and retain reasonable doubt.
With no card house to build, the players each undergo an interview/interrogation to relate their side of the story prior to the trial. Obvious omissions, lies, contradictions, etc give the GM additional opportunities to restart the house of cards should it fall. If it cannot be built successfully at all, the case is thrown out of court.
The interviews follow the normal social combat rules for the system being used and if successful, the players can invoke a time limit on each card placement made by the GM when building the house. The more of them who handle the interview successfully, the shorter the time the GM has to build the house.
The GM can decide to stop building the house at any time, but the players can barter or make deals with the GM/Prosecution in order to force one more attempted layer to the house in exchange for a bonus "point" they will need to overcome in the trial. The GM can spend some accrued points to create a larger base if that is deemed valuable.
Phase Two: Arguing the Case
As Gomad notes, the suggested use of the Jenga towers runs counter to the purpose to which they would be put, so I would say that pieces only be removed from a Tower when the opposing side scores a significant point which weakens the case of the other. If this is a classic structure of Judge, Jury, Defense, and Prosecution, the lawyers would focus on blowing holes in each others' cases, while the Judge would focus on keeping them both in line.
Each lawyer would be granted bonuses to rolls for solid information and evidence, provable in court, and with success, be able to force their opponent to draw a block from their tower. Depending on how you want to run things, this could be a long series of challenges, or the difference in rolls could be calculated with one block being drawn per point of difference.
The judge could require a block to be drawn for any infraction.
The Jury could serve as an additional resource for the legal teams. (see below)
The rolls of the legal teams could be bolstered in the form of physical evidence, additional witnesses brought to light, jury empathy, jury doubt, and convincing testimony. These areas are where the players of all the characters from accused to entirely innocent can contribute.
Action-oriented characters can be out racing against the clock to bring new evidence to light. Challenges can be for finding, reaching, and obtaining the evidence or witnesses, and then racing back for a dramatic revelation.
The talkers and faces of the group - especially those who enjoy the opportunity the interplay a trial can create - can take on the challenge of a non-verbal social combat with jury members as they try to sway them with their charm and obvious innocence from their varied places within the courtroom. As the trial goes on this extended challenge will rise and fall on the tide of the Judge's influence as well as the relative success of the lawyers.
This can be heightened by accidental or illicit interactions between a character and a jury member during a recess or other incident.
Jury members swayed to one side or another will reflect this in their posture, faces, etc and will thereby serve to affect further rolls and interactions and also set up the third phase.
Phase Three: The Jury Deliberates
This is where you can mix things up a bit. A PC can take on the role of a juror for every one swayed by the defense during the trial. This lets them have a real voice in the proceedings and gives them a direct channel toward earning their freedom. The GM will take on the role of the Jury foreperson, and the other jurors (which will need to have the majority) and the PCs will need to present the case evidence in their own way, seeking to figure out what will earn the votes of the largest number of jurors.
For the more action oriented of the group, this can incorporate flashbacks of 'how things really went down' complete with rolls and the like.
Using a deck of cards, jenga towers, and quickly intercut cut scenes of action and desperation with trial procedures and tension, plus a little bit of bidding and brinkmanship, you should be able to abstract the legal system into something quick, and engaging.