All rounder Räikkönen
In the past couple of years Kimi Räikkönen has raced snow mobiles (like your hero Gilles), saloon cars and boats. Any chance a guy like Kimi will do a one-off Le Mans, Sebring, or even Indy? Unlike other F1 drivers, he seems open to anything and nothing seems beneath him.
Kimi Räikkönen is indeed an unusually free spirit in today's rather regimented Formula 1 - as you point out, he won a speedboat race last year, wearing a gorilla suit and entered in the name of ‘James Hunt'! Did he have permission from Jean Todt, in advance, to take part? I somewhat doubt it...
However, events of that kind are one thing, other races - like Sebring, Le Mans or Indy - quite another. Ferrari, I'm told, pays Kimi in the region of $45m a year, and you may be very sure that precludes his racing in anything other than F1. If he were to drive at, say, Le Mans, and broke a wrist or whatever, that would keep him out of several Grands Prix, and no modern F1 team would risk that.
Drivers like Jim Clark and Graham Hill, as you say, took part in all kinds of races, but they didn't earn very much from their F1 contracts, and there were far fewer World Championship Grands Prix. I don't think it's that today's drivers would be afraid of racing anything else - more that their ultra-lucrative F1 contracts don't allow it.
Incidentally, much as it saddens me to say this, importance of the Indy 500 is so much reduced - certainly in European eyes - these days that I think it's highly unlikely any F1 driver would be interested in taking part in it. When Clark and Hill and others went there in the '60s, it was the biggest race in the world, they were taking on the likes of Foyt and Parnelli Jones - and the money on offer dwarfed anything available in F1. Different days.