Why is the British royal family always one step ahead of anti-monarchists and pro-republic people ?

Well I'd like to think they're survivors. To add to what to Pheemz2 was saying, not only can they trace their lineage all the way back to the house of Wessex, there is evidence that the House of Wessex was founded by someone who was himself half Romano-British. This means their family have been monarchs or chieftains in these isles for well over 1400 years, perhaps even dating back to before the iron age. Every time a monarch has been deposed or civil war has broken out the institution and broadly the same family has survived. If the ability to detect and avoid traitorous plots could evolve, this may well be evidence of it.