It is commonly assumed that, had he not died, emperor Frederick III of the German Empire would have liberalised Germany and possibly avoided WWI, potentially paving the way for a survival of the German Monarchy. How likely would this actually have been? Was he truly a liberal?

There are a few problems with your counterfactual.

The first is that Kaiser Frederick III had so little time to leave any track record of what he would actually DO as kaiser (rather than just talk about), and that so much time elapses between his death and the onset of the Great War, that your counterfactual is basically impossible to refute or to confirm to any reasonable degree. There are simply too many unknowns and too many variables to try and account for. Whether or not you present it as "True" (aka "Likely") or not merely comes down to if you WANT to believe it or not.

The second is that the baseline elements of European politics are unlikely to change significantly no matter who is Kaiser. France is still going to be courting the strongest allies against Germany it can find. The Balkans are still going to fall apart. Russia is still going to ram heads against Austria-Hungary over the Balkans as the Ottoman Empire weakens. Austria-Hungary is still going to be hamstrung by the 1867 Ausgleich. Bismark is still going to get really old and have to retire, and no matter who is Chancellor after him someone in Germany is going to have to choose to back Russia or Austria, because that relationship structure was always untenable in the long run. The great majority of the underlying causes of the Great War are MOST LIKELY going to occur no matter who is Kaiser and no matter what government structure Germany has.

The third is that you assume Germany wasn't a increasingly a liberal environment anyway. Did Germany have a Kaiser? Yes, but the Kaiser's power was not unlimited. Kaiser Wilhelm II already governed by largely doing what his cabinet wanted to do anyway (and they generally just ignored him when he tried to interfere), so a British style "Consent of the Cabinet" executive would have been functionally little different. Germany had universal male suffrage (the the Reichstag, but not Bundesrat) and a robust parliament with budgetary control. Germany had a powerful Socialist party (which held 110 of the 397 seats of the Reichstag in the 1912 election, the single largest party) that was stridently anti-war. Germany had the most progressive social welfare programs in Europe (and probably the world) at the time as a result. The trend in Germany was inexorably toward a stronger and stronger parliament. During the Zabern Affair in 1913, the Reichstag nearly compelled Chancellor Bethmann Hollweg to resign, even though that was not technically within their power (opinions on this differ, though Charles Emmerson presents it in this light in 1913: In Search of the World Before the Great War.) Nor was Germany nearly as military obsessed as some would have you believe. According to Max-Stephan Shulze in The Economics of World War I, Germany spent less per capita on defense than France or Britain as of 1913 (which had been true since at least 1905). They had a premier military, yes, but Germany was also the beacon of science and philosophy in 1913, and they were proud of it.

For my part, I don't dig the hypothetical. The Great War was not an event driven by the Kaiser or any sort of autocratic government. Far too many of the inputs that led to the Great War are MOST PROBABLY going to occur anyway no matter who is Kaiser and how much power the Reichstag/Bundesrat have (the German socialist party did vote unanimously in support of the war, after all). Of course, different people will react to crisis points differently, and it took several crisis points to get to the Great War, and if the Chancellor isn't von Bulow then Hollweg, or if the Secretary for Foreign Affairs is someone different then they make different choices... so we'll never know.