The Hapsburgs fought the French for a long time why couldn't they successfully put on their own on the throne in Paris?
Because they kept losing every time they went to French soil. Also post reformation, their ability to draw manpower from the Holy Roman Empire was weakened.
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The Hapsburgs had the money, the soldiers, and the clout. If anyone was placed at that time to re-create the borders of the Roman Empire and more.
Not seeing it happen. A lot of the funds were sucked away fighting the 30 Years War and on and off wars with the Ottomans. In effect, the Hapsburgs had to protect all of Europe from the Ottoman ambitions, while everyone had their own agenda. So much for the money. Keep in mind that when there was an Anglo-Austrian alliance, the English and later British kept pouring money into the throat of the Hapsburgs. Not saying they were terminally weakened at the point, far from it, but they would not have the ability to wage war after war to recreate the boarders. It might be find pretext (harder than you might think), go to war, win, consolidate for 3 decades, and then repeat. All while keeping the Ottomans out of their Hungarian lands.
Also note that France actually allied with the Ottomans at one point to destroy Hapsburg hegemony. This failed. It simply didn't occur to the French that if it worked, the Ottomans would likely treat them the same as other non Muslim states they signed agreements with... destroy them once their use is over.
Why couldn't Britain fall?
Oh I don't know, maybe you need control over Normandy to launch an invasion of England without massive numerical superiority? And France is kind of sitting on that land. And the best way the Hapsburgs could beat the French would be using the English to augment Austrian and Spanish forces? The massive Spanish Armada might have been able to defeat the English (England was in a sorry state at that time, if it's time to invade from a farther port, it's from Henry VIII's life to the beginning of Elizabeth's reign), but that POD means the reformation is strong in the Netherlands, Saxony, Denmark and the Palatinate, meaning even a Hapsburg in the 30 years war is likely to be long and costly.
A Personal Union with the Austrian Hapsburgs and the English might work and then worm its way to some French conquests. The problem is... well how are you going to do that?
Not that Britain won't fall, but the most likely point is too late to prevent a ruinous war