This may or may not be true and probably would not have effected the outcome but no harm making a note to self not to close a flight plan until safely on the ground?
For almost all countries outside the UK the answer is yes. If you don't close the flightplan, they come looking for you. On a controlled field, your plan is closed for you upon landing automatically. For an uncontrolled field, you either close by phone after landing, by calling a designated number, or you close the plan when signing off with the last "formal" ATC unit and when in sight of the field. Presumably, in that last case, if something happens after closing, somebody at the field will have you in sight, hopefully, and raise the alarm.
In the UK the situation is very different though. An overdue VFR flight plan, to the best of my knowledge, is NOT used as trigger for a SAR operation. Instead you have to designate a "responsible person" (a friend, but ATC units offer this service too) who you let know of your departure and your safe arrival. If that second phonecall is overdue, it's the responsible person who initiates the SAR operation. All this is completely independent of your flight plan and because of this, in the UK, flight plans are never "closed".
In any case, being overdue with closing your flight plan or failing to notify the responsible person will not have the helicopters scrambling straight away. When you're more than half an hour overdue from your last known ETA, they first start calling the destination and any alternates you have filed. They will also try to contact you via your contact number (mobile phone) on the flightplan and the operator/owner of the plane. And only when a phone search doesn't turn up anything (and this may take up to half an hour) will the helicopters start flying.
So the earliest that somebody will come and actually look for you is probably at least an hour from the moment your ETA expires. Of course, that may be already several hours after you've had the accident, and even then it might take several hours (days, or even years in some cases) for the rescue services to find you. The more accurate the route in your flightplan was listed and flown, the higher the chance that they find you reasonably fast.
Staying in contact with ATC (in the UK formally called "alerting service", which is implicit in a FIS) and, if possible, declaring a mayday early on will have much more immediate effect than a flight plan in any case.