We generally cut "wood' on the table saw, although I have cut aluminum and acrylic plastic, and Lexan.
There are 2 kinds of wood, man made and lumber from trees
Man made, plywood, MDF and other composites are the safest to cut, since there are no internal stresses. Lumber from trees does have internal stresses, some woods more than others and some grain directions a lot more than others, depending on where in the tree the wood came from.
Ripping or cross cutting man made wood is all the same as far as the physics of the wood, it doesn't care which way it's being cut. Yes, there will some tear out cutting across the grain on plywood, but the correct blade will take care of that issue.
Ripping, cutting down the length parallel with the grain for the most part, using the rip fence is what the table saw does better than any other tool with a circular blade. And they have been doin' it for many years. When ripping lumber from trees,
the internal stresses can close the kerf behind the blade and it will ride up at the rear,since that's the rotation direction and KICKBACK. Not so with plywood, no internal stresses.
Kickback on plywood occurs when the workpiece is not held tightly against the fence , either because of insufficient bearing surface or operator error or both. The use of a splitter or riving knife will all but eliminate this issue if proper technique is followed....forward pressure, downward pressure and pressure inward towards the fence.
Crosscuting using a miter gauge,( mine has an backer extending out on both sides of the blade above) or a crosscut sled is the safest way to deal with shorter workpieces. Some miter gauges have clamp to hold the work securely. I used to use mine, but not any more. Crosscutting using the miter gauge, even with an entended backer board has it's limitations when cutting longer (more than 6 feet) pieces, since any "wiggle" out at the end translates to a lot more at the blade end resulting in an inaccurate cut. Simple physics.
When possible I crosscut my work on the RAS or the sliding miter saw, but I won't hesitate to use the miter gauge on the table saw.
For larger/wider panels (more than 20" or so) the miter gauge become useless since it won't reach, you then use the fence, with proper technique...as above. So, sheet goods or panel work is a little different than lumber from trees, mainly in that lumber is a lot thicker...up to 3" maximum, one pass on a 10" table saw. ...and not every 10" table saw will make that cut in one pass, but that's another issue entirely....blade, horse power, etc.
This may be info already known to some, but there are newbies who should understand that there is a lot of simple physics involved in woodworking. Hope this helps!