There are several reasons why you should marinate before cooking:
Many marinades contain raw ingredients that should be cooked along with the food being marinated, such as garlic or ginger. In some cases this may actually be a health hazard (raw garlic can harbor botulism), in other cases you'll simply end up with an undesirable pungent flavour.
Many marinades also use some amount of acid, which helps to kick-start the process of breaking down the connective tissue and tenderizing the meat. This is an especially big deal for chicken breast meat and tougher cuts of beef or pork, because most cooking methods (except for slow-cooking) don't do much of this and tend to produce a tougher, chewier end result. Once the meat has already been cooked, using an acid-based marinade won't give you much more tenderization; you need the combination of acid and heat.
Finally, many spices release most of their flavour when heated (cooked). Examples are cinnamon, saffron, star anise, and various types of hot pepper or chili powder. If you marinate after cooking, you won't get much flavour from these.
Of course if you are using a very simple marinade such as soy sauce and are marinating only for flavour purposes, then it might not matter much, but that's the exception rather than the rule. Marinating afterward can certainly improve the taste, but most of the time you'll end up with better results by marinating beforehand.
The only method I have personally found to be reliable for grilling/pan-frying chicken breasts to a relatively uniform doneness is to pound them very, very thin with a mallet or rolling pin. Thin, as in scaloppine-thin, so that it cooks almost instantly in the pan.
Every other stovetop-only method is almost certainly going to produce a bland, tough cut, regardless of whether you press it down or not. As noted in my comment, my usual (lazier) method that does not involve pounding is to get a nice sear in the pan, then jam in a temperature probe and bake it in the oven until it's done (the USDA recommends 165° F, I usually don't go quite that high).
If pressing the meat actually accomplishes anything at all, it would most likely be to just squeeze out whatever tiny amount of precious juices the breast does have, and possibly give you slightly more even cooking on the exterior only; it will not help to cook the interior much faster unless, as stated above, the cut has been pounded extremely thin and flat, at which point it doesn't really matter.
P.S. Salt and olive oil is a terrible "marinade" for any cut of meat, especially a chicken breast. The salt is just going to get suspended in the oil and never reach the meat at all, and the oil itself won't have much of an effect on such a lean cut. You really need to change your marinade as well, preferably to something water-based (or at least not 100% oil).
Best Answer
I'd say the difference between frying or even using a griddle pan is that compared to a grill [US] barbecue [UK] that the latter generates more heat all round the food as it's cooking, as opposed to mainly the portion touching the surface of the pan.
With that in mind, some possible solutions…
Fry then oven - would give you the surface texture plus enough time to cook the inside. [Oven then fry might work too, though the surface would then be more difficult to fry evenly].
Butterfly the chicken before marinating, or even just slice in half - half the thickness, quicker to cook to the centre, with double the crispy surface & more marinade on it. Stronger flavour & a change in the proportion of textures, but you may like it that way.
Broil [US] grill [UK] with a more controlled temperature. You could achieve a similar, though not identical result compared to the default grill/barbecue method. You can decrease cooking time slightly by using metal skewers through the centre, like a kebab.