If you can pass your base checks and medicals every six months then professionally you are only limited by the ANO of the country you're working in. Witness my friend Nigel Osborn (above) and myself and quite a few others out there. WELL past 60... Your insurance company are of course lying to you as usual.
However as you rightly say, once you've passed the age of late 50's it's not easy to get full-time work; but contracting can be rewarding and more flexible for those with other things to do in their lives. And frankly if you still rely totally on flying to make a living at this age then consider this:
FACT no.1: Helicopters fulfil a need; they don't really provide much of a financial return on their operations and therefore don't usually attract the most capable senior managers.
FACT no.2: Helicopter companies tend to employ large capital assets and have high cash flows. This makes them "interesting" to the sharks of the business world.
FACT no.3: Helicopter pilots and engineers are therefore working in an unstable industry and must build up some form of second financial interest to safeguard their future out of the money they earn while they are flying.
Recommendation: NEVER trust your employer to have your best interests at heart. ALWAYS have a fallback plan (same as in the cockpit or the hangar).
Sorry if this is a downer but most of us are in the helicopter business because we like it; that does not make it good for our financial health.
So I suggest that you use your training qualifications and go contracting wherever you can in the world if you MUST still fly; if you have the option it's probably better to do something pre-planned or different.
Whatever you decide, good luck!