I don't think it'll ever be as great as people like you want it - Picard sets a high standard to live up to
Too true.
It would, however, require some very idealistic people to be consistently in power, and that's harder to do than it should be.
That's not entirely the case. I won't deny that individuals can sometimes massively alter the course of events - I do deny historicism. But democracies usually tend to operate in paradigms, sustained by uncodified doctrines and settlements, by broad consensus - and individuals are limited by that tendency. Once a policy is established it pertains to a greater or lesser degree as long as it is seen as successful: One victory can frame the debate for decades, making some positions acceptable and others completely bonkers (until a major failure causes a paradigm shift, breaking the previous consensus and allowing "the impossible to become inevitable" as Thatcher put it).
With a very few exceptions, individual leaders are limited by circumstances - the limits of the debate affect the platform they are elected on, that platform limits what they can reasonably do in office, the general trend of policy affects the composition of the legislature which in turns checks the executive's freedom of action (in parliamentary democracies, leaders who depart from a consensus run the risk of being removed by their own side; in presidential ones, they suffer gridlock). The Churchill who won in 1951 was an exceptionally different politician from the one who lost in 1945; the Nixon who won in 1968 very different from the one who lost in 1960.
So for instance, a successful Wallace Presidency with his re-election in 1948 might lead to Stevenson or Kefauver or William O. Douglas as President in the 50s; or possibly it makes Taft a more plausible candidate. But I think it's more likely to affect how Eisenhower runs and wins, to force adaptions in the Republican platform (and compare the hawkish GOP platform of 1952 with earlier isolationism and you can see how circumstances affect policy). An Eisenhower Presidency that followed a successful Wallace Presidency would be very different from the Eisenhower Presidency that followed the successful Truman Presidency in OTL. Of course, there are only a few points where these paradigm shifts can take place - although, in AH terms, smaller shifts can go on to create ones that weren't there in OTL.