I think if you fail a raw bench at the bottom it's because you didn't stay tight enough. Perhaps the:
- Chest flattened out
- Upper back flattened out
- Lats didn't stay flared
- Elbows splayed out
- Lost your leg drive
Or you don't know how to bench to take advantage of the above points. If you can maintain the above positions you shouldn't really fail at the bottom, it would be closer to the mid-point. Once you're confident that you can actually Bench in a way which utilises the above list of points there are further ways in which you can strengthen those muscle groups.
Supramaximal (above your max) weights are enable you to practice that body position with heavier weights. This could be in the form of board work, reverse band work or slingshot work or even a combination of the above. For me I regularly handle 180kg+ in training and have on occasion done 220kg, so being able to manhandle 140 raw isn't a big deal for me because it usually feels pretty light in my hands.
Paused work is an excellent test of how well you can keep tight under tension. It is a brute force approach to building the stabilisation needed for heavy weights. Paused work is great, paused work for higher reps is even better as your stabilisation muscles are under tension for longer. Paused work for higher reps with feet on the bench is perhaps the roughest form of benching to improve your upper back/shoulder/elbow stabilisation for the Bench. If you can keep a good back position with no help from your legs, for a high rep set and with a decent weight you are making some strides towards increasing stabilisation.
Strengthening the upper back is a massively important area. The ability to keep your upper back/thoracic region arched and shoulders shrugged back is absolutely vital to a raw bench. There are disagreements about whether elbows should be tucked or lats used comparing raw and equipped benches, but there is NEVER any argument about upper back involvement in a raw bench. It HAS to stay tight and shrugged back and your thoracic back HAS to stay arched. If you lose either of these positions or you never took advantage of them in the first place you WILL limit your benching potential.
In further addition strong legs/back will provide a strong base of support for your press. If you can't maintain your leg drive, and the position of driving your upper back into the bench then you'll lose tightness as well. All of the above points usually interconnect, that is if something fails then something else usually follows suit, if your upper back flattens out for example usually the chest caves in and the leg drive is mute. So it's absolutely vital that you practice the Bench over and over so you can get it right every time, in addition to that employing assistance to strengthen areas listed above will help only if you actually remember to use them during the lift.
This is only really a start in terms of bench advice and things I would do/have done to fix my own lagging bench. I could go on about assistance exercises, paused work, specific chest work, strengthening a particular groove etc.
But for now, hope that helps.