You have several options for such operations:
1) Use a feather board just ahead of the blade (the feather board should end just a bit — a quarter to half inch — before you reach the first leading tooth of your blade. A second vertically mounted featherboard (secured to fence) beside or behind the blade can also increase control if you do not have an off-cut table to catch the piece or if the board is very long by holding the board securely against the table can improve control and helps to resist any impending kickback event by holding the wood down and forcing the rear teeth to cut (if they must) rather than lifting and throwing the board — but you will need a push stick or sacrificial piece of wood to finish pushing the board through and past the blade and feather board.
2) Use a pair of push sticks to help apply some pressure towards the fence while also pushing the board through and past the blade (but remove or relocate the lateral push stick before you reach the blade as you never want lateral pressure against the blade or you risk a kickback.
3) Use a bandsaw, there are many tablesaw operations that are safer on a bandsaw if your bandsaw is large enough to handle the workpiece since there is no kickback risk, though you must still be concious of the blade itself, but a sacrificial block of wood can be used to safely push the workpiece past the blade if too narrow a cut fir a push stick and a pushblock, or a second block of wood, can keep the board pressed tightly against the fence keeping hands away from the blade.
4) Use a jointer or thickness planer to remove the excess — you can usually remove up to 1/8th inch in a single pass on the narrow edge and perhaps a bit less if working the broad face, depending upon your equipment and the difficulty of the wood itself (though a jointer should be used if removing wood from the narrow edge as a board much taller than the edge being jointed/planed is not guaranteed to be stable on a planer whereas jointers are meant for such).
5) Use a specialized pushblock like the Gripper you included in your post.
6) Accept that sometimes the safety guard must be removed from the tablesaw for certain cuts while exercising proper safety — use of push sticks, push blocks, sacrificial wooden push stick/board (for narrow pieces), feather boards, splitter or riving knife, proper stance to side of board (stay out of kickback danger zone), etc.
7) Buy or engineer a jig or tool to make an operation safer, easier, or more efficient — a time honored tradition. For example, for short pieces a miter sled or specialized miter gauge might make an operation safer by allowing clamping or other support that keeps one’s hands well away from the blade.
8) Set blade height to no more than 1/4” to 3/8”, typically the height of the tooth gullet, above the height of the workpiece you are cutting so as to reduce the amount of exposed blade for through cuts.
9) And before any power tool operation, despite safety equipment, always think through the operation and ask oneself what could possibly go wrong and what has one done to mitigate that risk in terms of technique, safety equipment, or even the operation as a whole (e.g. would a different tool or approach be safer?). Thinking through the physics of the intended operation and how the tool and wood might react to different scenarios that may play out is the most critical safety step.
The above are all pretty generic advice for tablesaw safety, but they will usually go a long ways towards keeping one safe — especially the last step that undoubtedly led to you asking this question. Just be careful to always think things through as sometimes safety equipment can lead to a very false sense of security — for example, if you are pushing hard with a push stick or pushblock and it slips (especially if your hand is otherwise in mid air) where will your hand go in the fraction of a second before you can react and catch yourself and will your safety gear allow an adequate reaction time. It’s ok to sacrifice push sticks and pushblocks if things go wrong, that is what they exist for, but you always want to ensure they are all that gets sacrificed aside from the board itself!
