Personally, I don't think seeing the same character the exact same way across several films is particularly interesting.
I see nothing wrong with changing up the character, showing new dimensions or having the character grow across (or even between) the films.
It wouldn't have made any sense to see the "Raiders" Indy in a sequel set more than twenty years later.
However, I agree with Matt deMille's suggestion that the softening of Indy in KOTCS was due more to a stylistic choice by the filmmakers.
It's not really Indy that's gone soft in KOTCS (and, to a lesser extent, LC), it's the movies themselves that are softer - much less gritty, bloody and suspenseful.
The tone of LC was much lighter than its predecessors, with much more comedy and with less intense action sequences. I think LC is very enjoyable with a lot of good qualities, but, as an action movie, it's tonally different than its predecessors and is, at least partially, lacking the earlier films sense of urgency and danger.
Unfortunately, those cartoonish qualities were amplified tenfold in KOTCS, which is an utterly bloodless affair. Indy doesn't kill anyone. He only really gets into one serious fight. There was - for me, anyway - NO sense of actual threat or suspense anywhere in the movie. Russian soldiers fire machine guns at our heroes when they're five feet away and hit nothing but leaves. Indy - Indy in his fifties, mind you - crashes through glass and metal comfortably, without skipping a beat. None of the other films were that deliberately preposterous.
I don't think the movie actually bothered exploring in any meaningful way what it would REALLY be like to be Indiana Jones fast approaching 60. What was endearing about the original "Raiders" Indy was his fallibility. Sure, he was heroic and tough and relentless, but he was far from a superman.
With the exception of a few seconds in that aforementioned Dovchenko fight, I never felt Indy was in any real danger in KOTCS and I never felt like he was up against odds he couldn't handle. In the first two movies, we see Indy barely struggling to stay ahead of the game.
It would have been great if Indy's age and weaknesses were made a part of his character arc. If we did see a changed Indy in mind and body and if those implications were explored in the film's story and action sequences. What better way to make Indy sympathetic than have him be an old guy who isn't quite up to the challenge like he used to be.
Other movies have flirted with similar themes - and have also failed to deliver. In "Lethal Weapon 4", Riggs is faced with the prospect of getting old for the first time and worries he's no match for the incredibly young and powerful Jet Li. That is, until the very end, when he manages to win the fight, anyway.
In the James Bond film, "The World is Not Enough", the filmmakers set up a potentially amazing dynamic between an injured Bond - who has to bluff his way back into active duty - and a villain who can feel no pain. Unfortunately, that wonderful concept was all but forgotten by the film's end.
Ultimately, KOTCS does definitely continues and greatly expands on the light tone introduced in LC, but it comes from behind the camera. Despite a few quips like "not as easy as it used to be," Indy's age and physicality isn't mined for any character-based or thematic resonance. If anything, Indy is too capable, too competent in the film. It's only the age of the filmmakers that slows him down.
P), but you give legitimate reasons and not just silly and irrelevant like Deckard was.