Has the office of the Prime Minister become 'Presidential'?

There is certainly a case to be made that in recent years, the office of the Prime Minister has become more 'Presidential'. One of the biggest reasons for this is the rise of new media. Due to the ubiquity of television news and internet streaming in the modern age, the Prime Minister (PM) has found it easier than ever to reach the masses as the spokesperson for all government affairs. The PM remains 'Primus inter pares' (first among equals) with the rest of their cabinet, de jure. However, de facto the Prime Minister has elevated them self to a position above the rest of the cabinet and their party, adopting a more 'Presidential' style of addressing the nation. We can see examples such as Theresa May's speech outside Downing Street after her election which set out the new government's agenda. The PM has also become more 'Presidential' due to the state of the modern world. The rise of international trade, treaties and political unions has meant that the PM is forced to become the international spokesperson for their nation, along with their use of the prerogative power that is signing international treaties, they act in much the same way that the US President does on the international stage. Finally, the PM has become more 'Presidential' in the way they act as the 'Chief Comforter.' This means that when a country faces national crisis, the Prime Minister is there to comfort the nation as the representative of the nation, taking much of the Crown's responsibility. One of the most striking examples of this was after the death of Princess Diana, in which PM Tony Blair addressed the media, and the nation, using the famous description of Diana as the "people's princess." In that moment he appeared to share much of the same pain that the nation was, in a similar way to to how the US President comforts the nation after a mass shooting or a terrorist attack.
However, in the de jure sense, the powers of the PM in no way resemble those of a President, and arguably have reduced in recent years. The PM still does not share the comfort of their employment in the same way that a President does. They are at all times vulnerable to a vote of no confidence from the House of Commons, whereas the removal process is much more difficult for a President (who must commit a crime to be removed.) Equally, if we observe Theresa May's time in office, we can see a rise in the voice of Cabinet who have taken no guilt in addressing the nation in their own way, even if it contradicts the Prime Minister. A good example of this is when Boris Johnson as Foreign Secretary set out his own Brexit plan in a popular broadsheet newspaper, completely rejecting cabinet collective responsibility. A President would not face such resistance from their cabinet as they have the comfort of being able to fire them regardless of the political climate. We have also seen in recent years that the Prime Minister has reduced their more 'Presidential' powers. Tony Blair's precedent of calling a vote before going into a major international conflict has meant that the PM can no longer operate their prerogative power of declaring war, something that a President holds in many cases. We saw this reduction in power in 2013 when David Cameron lost in the House of Commons on a vote that would have sent drone strikes into Libya. If he were a President he could have sent those strikes without the consent of the legislature.