With regards to cost of the chassis I don’t think that this is relevant. This is for two reasons –
firstly, that when the drivers see red in the cockpit, the last thing that is on their mind is the cost of the equipment around them. As far as I know, there are no contractual clauses that require them to personally fork out for the damage they cause (as that would be hugely unfair).
Secondly, the cost of the chassis is not worth as much as the cost of the prize at the end. i.e. if they risk say £100,000 for an accident, but by doing so overtake one car, this may give them an extra £5,000,000 at the end of the season. Breaking it down race by race, it will always be more beneficial to overtake a car at risk than it will to stay behind.
The only time this will not be the case is either when one team has won/solidified a lead in the championship that is not worth risking [in which case, point 1 about the racing instincts will overrule it] or when a team, such as HRT has only limited parts for their car and therefore cannot risk an overtake. However, the teams with limited parts tend to be racing at the back, therefore there isn’t much overtaking going on.
Fragility of the cars, however, is definitely a factor. Why risk your front wing at the first corner which will ruin your race? I don’t know how you could fix that – because the nature of open wheel racing means that the delicate parts of the car are exposed. Accordingly, the cars will always be fragile. To me, that’s what makes the sport great, so I wouldn’t want to change it.
Further, fragility is also not the only reason. As others have mentioned, circuit design, aerodynamics and tyres all play an equally large, if not greater role than the fragility of the cars.
To me though, there is no ‘problem’. And that is in itself, the problem with F1 and overtaking. The desire to have commercial excitement is what has driven this whole ‘f1 is boring’ attitude that is in itself the real reason for the wide publicity of a non existent problem. F1 has never been about lots of overtaking, its been about skill and those rare overtaking manoeuvres that take your breath away. Its about so much more than overtaking. The real problem, therefore, is the press and the commercial desire for a ‘perfect’ product. To me though, and many F1 fans, F1 has been pretty damn perfect for a while.