Pitt the Elder is counted as Prime Minister, but that is at least a little anachronistic. In the 1770s Lord North was still denying that the office existed.
Though its traditional to count Walpole as the First Prime Minister, a case can be made that the office as its understood really didn't exist until North persuaded George III after Yorktown that he had lost the confidence of the House of Commons and that this meant he had to resign. Until then the "Prime Ministers" were just the dominant personalities among the ministers. Afterwards the Prime Minister became the main link between the sovereign and parliament.
In any case, I think the tradition of counting Pitt the Elder as a Prime Minister is incorrect. The office of "Prime Minister" did not exist in any legal documents until the twentieth century. It is associated with the office of the First Lord of the Treasury, the closest successor to the medieval Lord High Treasurer.
In 1766 Pitt the Elder was made Lord Privy Seal and Earl of Chatham. He was not the First Lord of the Treasury, the Duke of Grafton was. Within less than a year Chatham had what seems to have been a nervous breakdown and took no further part in government affairs. He didn't resign as Lord Privy Seal until 1768. Historians count Chatham as Prime Minister between 1766 and 1768, and then Grafton as Prime Minister between 1768 and 1770. This is despite Chatham not being First Lord of the Treasury at any point during his "ministry" and not doing any government work during most of it. Though everyone assumed Pitt would be the dominant minister in 1766, conventions had not hardened and in retrospect it would be more consistent to count the Duke of Grafton as Prime Minister during the entire four years.
In 1746, George II agreed to bring back the "Broad Bottom Ministry" with Henry Pelham, Duke of Newcastle, as First Lord of the Treasury and reluctantly agreed that Pitt the Elder would be Paymaster of the Forces. The leading government minister in the Commons was Henry Fox, the father of Charles Fox. In 1754 Henry Pelham died as was succeeded as Duke of Newcastle and First Lord of the Treasury by his brother, Thomas. Pitt the Elder was dropped from the Cabinet after a year. After military reverses for the British in the Seven Years War, the ministry fell and Henry Cavendish, Duke of Devonshire, became First Lord of the Treasury, and Pitt the Elder was brought in as Secretary of State of the Southern Department (very loosely the antecedent to the Foreign Ministry) and the leading minister in the Commons. Pitt the Elder dominated this ministry and the subsequent ministry formed in 1757 with Thomas Pelham, Duke of Newcastle as First Lord of the Treasury and Pitt still as Secretary of the Southern Department and leading minister in the Commons and the dominant personality in the ministry.
George III became King in 1760 and clashed with Pitt the Elder almost immediately. The latter resigned in 1761, but Newcastle remained as First Lord of the Treasury until 1762 with Henry Fox again the leading government minister in the Commons. George III then made his private tutor Prime Minister, which itself should indicate how similar that office was from what it later became.
Since North resigned in 1782, there has been only one case where the Prime Minister was not also First Lord of the Treasury. Robert Cecil, Marquis of Salisbury, is counted as Prime Minister in 1885-86, 1886-92, and 1895-1902. Salisbury was only First Lord of the Treasury during a few months after these periods, in 1886-87, and did not live at 10 Downing Street. He was instead Foreign Secretary, except in 1900-02 when he was Lord Privy Seal, as well as the government leader in the House of Lords. He never campaigned in any general election for his own government. If you view the Foreign Office as the successor of the Southern Department (the Northern Department and Southern Department were reorganized into the Home Office and Foreign Office in 1783), these were the same positions Pitt the Elder held in 1757-61 and 1766-68. In 1885 the Conservative leader in the House of Commons, Stafford Northcote, was made First Lord of the Treasury but also made a peer. For the rest of the Salisbury governments, the position of First Lord of the Treasury was held by the government leader in the House of Commons, WH Smith between 1887 and 1891, and Arthur Balfour in 1891-92 and again in 1895-1900. Given the increasing amount of government business Arthur Balfour handled for his uncle after 1895, there is a case for dating his own government back to that year or 1900 instead of the traditional date of 1902.
After the Parliament Act of 1911, it became near automatic that the leader for the government in the House of Commons (usually but not always the leader of the largest party in the Commons) would also be both First Lord of the Treasury and Prime Minister. Between 1782 and 1911, the Prime Minister could be the leader for the government in either the Commons or Lords. Before 1782 the Prime Minister was pretty much just the dominant personality in the cabinet and the favorite of the King.
A distinction should at least be made between modern Prime Minsters, who are in office due to their being able to get Parliament to support them, and the pre-1782 early Prime Ministers where ability to manage parliament was one consideration out of several. This would make the first modern Prime Minister either North's successor Charles Watson-Wentworh, the Marquis of Rockingham, or given the short-lived nature of the three ministries in 1782 and 1783 and the fact that the First Lord of the Treasury was still not necessarily the dominant minister, Pitt the Younger who became Prime Minister in December 1783.
After the Parliament Act of 1911 the office of Prime Minister increasingly existed in statute (first mentioned in 1917) and who holds the office is straightforward. Between 1783 and 1911 the Prime Minister will always be either the Leader of the House of Commons or the Leader of the House of Lords. Its a matter for historians to determine which of the two was more important. The first time a Prime Minster sat in the Commons and was not the Leader of the House of Commons was in 1918. Before 1783 who was Prime Minister was up for debate and the entire concept of having a "Prime Minister" as head of government was considered to be an infringement on the constitutional powers of the King.
All that having been written, according to historical convention, William Pitt the Elder is counted as Prime Minister between 1766 and 1768, and not between 1757 and 1761. This is backwards. He never held the office of First Lord of the Treasury during either period, or at any other time in his life. Between 1757 and 1761 he was both the government leader in the House of Commons and obviously the dominant and most indispensable minister, until Britain had clearly won the war. Between 1766 and 1768 he was government leader in neither house, held a less important ministry, and for most of that period took little part in government business. Obviously Pitt the Elder should be credited as being Prime Minister during 1757 and 1761 when he was very effective, or never at all.
The actual candidate for "Prime Minister" between 1766 and 1768 are Augustus Fitzroy, Duke of Grafton First Lord of the Treasury and Leader of the House of Lords (most likely), or Henry Conway, Secretary of the Southern Department and Leader of the House of Commons, though William Petty, Earl of Shelburne and Secretary of the Northern Department is also a possibility as being closest to a dominant minister.
And to improve Pitt the Elder's effectiveness after 1766, improve his health.