I don’t think FRC needs a 4 speed transmission.
Based on the modeling work I did with shifting transmissions, I found that it was only beneficial to upshift automatically for improved acceleration if the low gear was very high, around 8fps for a 4 CIM 150lb robot, and this drove a desired ratio spread of <2:1 to put the high gear around 15fps. All of this resulted (in simulation) in optimum accel times, time to distance (short and long), and battery usage for pick and place maneuvers (which can increasingly use ‘low’ as the gear for short distance maneuvers).
However, for a 4 CIM drive, 8fps is not traction limited at 40a/motor with any reasonably grippy wheel. So that drove the desire for a 3 speed transmission, where 2-3 are used for ‘normal’ play and 1 is used for pushing.
Long ago, when motor power was significantly limited, shifting up sequentially would improve performance a lot (compared to throwing more power at it, which is common now). When the 33 4-speed was designed, this was the case.
Using dog and mesh shifting methods, it’s easier to combine 2 2-speed shifters with a different spread in series than it is to build a native 3 speed (with ball shifting a 3 speed is trivial to design, as you just add 1 more of the shifted gear sets). However, this results in a shift combination of L-L, L-H, H-L, H-H, where the 2<->3 shift is a ‘double swap’ shift. Automotive transmission engineers know that a double swap shift (more than one apply and release element) is a bad thing, and is extremely hard to synchronize properly. This can lead to temporarily being in a different gear (e.g. if shifting from 2-3 you can end up in 1 momentarily) which is really bad. So the answer then is to jump-shift, going 1-2-4 or 1-3-4. Then you get into some rather cool software, and a lot of design work to optimize all of the gears to get the performance benefit you are looking for. Definitely a cool project if you want an engineering challenge, but probably not worth it for an FRC robot.
So, basically, you go back and question why you wanted 3 speeds instead of 2 (or 1).
Now for some insight on automatic transmissions in cars. Modern automatic transmissions usually fall into 2 categories: Planetary wet clutch and automated manual/dry clutch.
Automated manuals are basically a series of dogs (or synchronizers, which are more complex dogs) each with 2 ‘gears’ and capable of shifting between 1, 2, or neutral. With many of these in parallel, any individual gear can be selected (1 at a time). The input is also coupled to the engine via a clutch, which can be released during a shift. Dual clutch transmissions have two of these pathways with separate clutches, and can ‘preselect’ a different gear on the other pathway then engage it by controlling the release of one clutch and the apply of the other.
Planetary wet clutch transmissions are basically a combination of planetary gearsets and clutches that either lockup (lock a segment to ‘ground’), input (lock a segment to the engine/motor), or connect 2 segments together. Virtually always, this is done with wet multiplate clutches, which can be slipped to smooth the shift. If you want to look up some cool gear layouts, look for Simpson or Ravigneaux type compound planetaries, those are basic 3+R and 4+R designs.