One of the odd things about melee is that STR affects not only the damage you do via the DM to the damage dice, but also the chance to hit (and therefore the extra damage you do with effect). Traveller theoretically moved from armour makes you harder to hit to armour reduces the damage you take if you are hit, but the STR mods for melee subvert that.
Personally I think your chance to hit and any effect mods should be skill plus DEX DM only, STR should only be added to the damage roll itself.
Whilst sword fighting (for example) does normally use most of the body (you are not standing square on trading blows like the automata on a clock tower banging on the bell), some blows do just come from the arm (snap cuts, cuts in retreat, stop thrusts etc.). Using the gross muscles allows power to be developed while the arm directs that power. If the arm were exceptionally strong there is no mechanical reason why the power could not be developed there instead (and the rest of the body providing a greater proportion of directionality than would otherwise be the case).
I was taught that climbing mostly used the legs for preference (in fact in most activity using the leg muscles is preferable to using weaker arm muscles). Grip is clearly important, but that is just a specific use case of "hanging on". Since there are one-armed climbers however it is clearly not an absolute that both arms need to have similar strength. The problem is that in MGT2 climbing is not STR based skill it is Athletics(DEX). This could represent being coordinated and not falling off (which is usually more important than being able to hang on for extended periods). As with all DMs for characteristics however this will be a Referee call. I could certainly see where STR was more relevant (a speed climb for example) or even END (for a mountaineering attempt) were more important. It depends on your interpretation of what the different stats represent (climbers and marathon runners are generally are wiry rather than bulky like weight lifters, but they are undoubtedly stronger than your average couch potato).
Another question would be what would happen if you had both arms replaced with combat arms? Two advanced arms giving you STR 18 (STR DM+4) would only be KCr150. This is a lot of money, but a fraction of the cost of other augmentations that provide much less effect but affect core STR (even if you pay premiums to make the combat arm natural looking etc.).