As you're posting in the powerlifting section then approaching it from a powerlifter perspective...does it matter where you feel it? Surely the aim is to lift the maximum amount you can with the least risk of injury, regardless of whether you feel it more in your chest or tri's or delts?
Unless you feel that this increased activation/feeling it in your chest more is going to help you lift more, in which case it does come down to your question of is the slight loss of stability worth it.
That is why I mentioned that I meant raw because you want to feel your pecs more on raw bench vs shirted being more triceps focused. I should add that I am talking about only implementing this during training, so I feel if it makes you a bit weaker or less stable during that workout, but leads to PR's it would be worth it.
Feeling wise is kind of important for better activation. For instance as I mentioned my thinking is the same as stretching your hip flexors to better activate/feel your glutes on squats. The pecs are the biggest agonist muscles in the bench press, so if you can train yourself to use/feel your pecs more in the proper setup I think it will lead to PR's.
I get you meant raw, I've never lifted in gear in my life so I'm coming from the same perspective. My question still stands though, basically, do you want a bigger bench or a bigger chest? These may be synonymous but not necessarily. There's a reason a some people don't build a big chest doing bench, it's because they don't use their chest as much. So reverse engineering that, would building a stronger/bigger chest increase their bench more than increasing their delt/tricep strength? It all depends on your benching style, dimensions, strengths and weaknesses. For example to improve my bench, I need stronger triceps not a stronger chest, as at the moment my lockout is the weakest and I never struggle off my chest.
I would disagree with the automatic assumption that you want to feel your pecs more on raw bench than any other muscle, at least as a generalisation! I have no clue your benching style/weaknesses etc.
As for the actual question I suppose it's up to you how much more vulnerable to injury it makes you and how much it would help you improve, then it's up to you to weight up the risks, no one else can really do that for you
I think of the stretching prior to training as a teaching tool more than a training tool I guess. As in for teaching the CNS vs focusing on training the chest, although it will cause the chest to get hit harder. It is exactly for the type of people you described, those who do not know how to utilize/feel their chest and only feel triceps and delts(btw these stretches also should help you feel anterior delts better). I do not mean they should feel it more than any other muscle, but I do think every raw bencher should be able to feel and fully utilize their chest to get the most out of their bench. I think this regardless of technique.
I also think it is an effective tool regardless of weakness, for instance someone who is weak at lockout, but is not fully utilizing their chest could gain a lot from learning to use their chest for 2 reasons. First being that the chest works through the whole range of motion, not just off the chest. Second being that since the chest can then do more of the work off of the chest it can save triceps and shoulders strength for lockout.
I was more of a triceps and delt bencher before I read a great article by Dan Green, he is a huge proponent of using chest for the bench press.
I would ask for the link to that article but pretty sure the admins wont allow it so I'll have a look around and see if I can find it. What you say is interesting though.
I do realise that the chest works through the whole range of motion, just more so at the bottom is what I meant, but your reasoning does make sense. I am by no means a good bencher so anything to help me get better is appreciated/of interest to me. Having said that as I do feel my chest working and as mentioned don't struggle off the chest it may not be as useful to me as to others, but still worth giving it a go.
Actually relating to the original message you posted, what stretches did you actually do? (It may sound stupid but I swear I'm really shit at stretching and when I do half the stretches I read about/get told about I feel nothing)
Seated Bent-over Infraspinatus Stretch and light Rear Deltoid Stretch. I usually never stretch unless it is corrective or weighted on off days.
Personally for benching I think the most beneficial thing is frequency. I only recently developed a reasoning for this and it is that unlike squats and deadlifts, bench press is not a natural human movement. For instance our ancestors used to always squat for sitting, pooping, etc. The hip hinge(deadlift) is a part of life for anytime they had to pick up something, an animal, rock, wood, etc. Besides maybe using a spear there was just not much horizontal pressing, so the CNS really needs to be trained for it.
I am by no means a good bencher, but by upping frequency I have seen much better progress.
*Edit: I would not recommend trying it just yet, I liked it a lot, but I would rather be a guinea pig to make sure it is safe. My chest and front delts are definitely more sore today, but also my shoulders feel a little rough. Although the roughness could be mainly from some overload work I did that day.