Although I usually prefer to expand existing skills rather than adding new ones, let me say that this thread has really good ideas, and yes Constitution and Strength would be nice to have more skills.
Stabilize Self (Con): this could be the easiest to do, since the psionic skill already exists with its own DCs; I think it would be nice to give it more uses tho, beside remaining conscious. Would you consider this skill (with a more general name in that case) to be used for other efforts of resistance to pain? I'm not sure what, but there must be something... only the death from massive damage comes to mind (already has a Fort save).
Endurance (Con): great idea, I think this sort of thing CAN be trained. The problem is that those situations always give a Fort save: if you replace the ST with an Endurance check, that can seriously hurt anyone who doesn't have this skills; if you stack the ranks with the ST modifier, that would instead sky-rocket the existing check. I'm not sure if it's nice, but maybe you could consider allowing both checks (I mean, one check with the best of the 2 modifiers).
Grapple (Str): this was discussed before in this forum, to make Grapple a separate skill on its own. Personally I think it would be great, because I see no reason why a good swordsman should be a good wrestler in a world where typically combat is fought with weapons. It would also make grapple a less cheesy tactic vs spellcasters, if every fighter has to pay half his skill points for it.
About Str-based games. Well, Roman-Greek traditions had many (you know Olimpic games, right?

) but they almost all already fall into BAB (javelin etc.), Jump, Swim or Grapple. The only one left which could be made into a skill is Run. I have a hard time thinking of a good way to do it however, because it may mess up with movement rules.
One last note: in this forum someone was writing a Str/Dex skill to replace BAB, which eah character would buy for single weapons (or weapon groups) separately. It would be another interesting variant, but I really think it's way too far from your idea here.