For the case of Bork, the GOP argument would only be valid if Democrats obstructed all of Reagan's nominees and if the obstruction was long party lines. However, neither of those are the case. Previously, Reagan's nomination of Sandra Day O'Connor passed the Senate with a 99-0 vote, his elevation of William Rehnquist passed 65-33, his nomination of Antonin Scalia passed 98-0, and finally Anthony Kennedy passed with 98-0. The vote for Bork was not rejected along party lines--6 Republicans voted against him, and 2 Democrats voted for him.
To the issue of Democrats getting rid of the federal judge filibuster at the end of 2013, the data shows that this was the result of an exploding number of filibusters by Republicans. The historical record shows that Republicans blocked the Obama agenda at every level of government, including their obstructionism to approve federal judges and other appointees. This pattern began shortly after Obama took office in 2009, continued throughout his presidency, of course culminating in the GOP refusal to allow Obama's SCOTUS pick of Merrick Garland to even come up for a vote after the death of Antonin Scalia.
All of this behavior was literally unprecedented. As the graph below shows, for most of the 1900s, the use of the filibuster was relatively low--typically less than 10 cloture motions filed per year, and many years there were none filed. This began to change in the 1970s. In 1975, the Senate made it easier for cloture to be filed--reducing the percentage from 2/3 (67 votes) to 3/5 (60 votes). The 1960s was a tumultuous era, the peak of the Civil Rights movement--the higher threshold for filibuster likely kept them to a minimum, but even still, there were very few filibusters during that decade.
In the late 1960s, Nixon & Reagan's "Southern Strategy" successfully changed the way that the public voted for Democrats and Republicans--the South transitioned from a Democrat stronghold to a Republican stronghold. Therefore, on the graph, I begin coloring the cloture motion bars blue (Democrat) or red (GOP) in 1968, but do not assign party prior to that (the 2001-2002 Senate is uncolored--control went back and forth due to a Senate tie). Since it is the minority party that filibusters, the color of the bars represent the minority party (ie, the 1969-1970 Senate was Democrat controlled--Republicans were therefore the minority, so the bar is colored red). The background color represents the party of the president (blue for Democrat, red for Republican). There s a gradual increase in filibusters from 1970s-early 2000s. The big jump first occurs in the late 2000s when Democrats were in control of the Senate at the end of the Bush years--conservative Republicans were filibustering Democratic Senate legislation, as well as opposing much of their own president's (Bush) agenda, such as the expansion of Medicare and immigration.
This significantly changed once Obama was elected. As the graph shows, Republicans as the minority in the Senate filibustered twice as much as had ever been seen before (2013-2014), eventually leading to Democrats getting rid of the filibuster for the one issue of allowing federal judges to be approved.