Lee's lack of a pardon was something of an administrative oversight combined with politics, but also maybe over exaggerated.

Per President Johnson's May 29, 1865 proclamation, Lee was required to separately apply for amnesty and citizenship. Lee complied with this, submitting his written application and oath of allegiance through General Grant in October of 1865. He in turn passed it on to Secretary of State Seward. Seward apparently had no desire to see Lee's status restored, so he allegedly separated the oath of allegiance and passed the it on to someone as a souvenir.

However, President Johnson issued a second proclamation on July 4, 1868, again pardoning participants and removing the requirement for the oath of allegiance. It was clarified further on December 25, 1868, to be unconditional and apply to all persons who fought for the Confederacy, as long as they weren't charged with some other treasonous act. The State Department still did not act to restore Lee's citizenship, and he died on October 12, 1870.

People wrote letters to the government constantly about Lee's pardon and citizenship, pointing out that they should be restored based on the above information. The Adjutant General's Office received so many petitions from the 1870s through 1975, that they created a separate file for petition requests and a form response letter in 1936. The letter stated, in brief, that the War Department agreed that Lee's citizenship had in effect been restored by the President's proclamation in 1868. All of these letters are available for viewing in the National Archives.

In 1970, an archivist at the National Archives found Lee's "misplaced" oath of allegiance. Senator Harry Byrd, a Democrat from Virginia, spearheaded the resolution to officially pardon Lee and restore his citizenship. It passed the House 407-10 five years later. Byrd took it up as a fellow Virginian and Democrat, but it would have been easily passed by anyone. It had nothing to do with the Southern Strategy. Lee was a very popular figure nationally in the 1970's thanks to the Lost Cause mythology. President Ford signed it as it was popular, and Presidents like doing popular things that make people happy.

Whether or not the 1975 resolution restoring Lee's citizenship and pardoning him was needed is a bit of a moot point now. Some, like the War Department, believed that Lee had already been pardoned in 1868. The grassroots movement gained traction because of the interest in Lee in the 1970s, the finding of the lost document, and because it was easy political points that cost nothing for politicians to act upon.

Sources: Emory, Robert E. Lee and the National Archives

/u/nilhaus covers this, but a couple additional points, including that one premise here is incorrect: Ford's actions weren't a pardon.

First, there does seem to be a bit of disagreement on why Lee's application was not in order. If you go back and look at the actual article that set off the 1970s movement - it's on a single page (181) in the Winter 1970 issue if you scroll down a bit - the details get significantly murkier in who was responsible for blackballing Lee.

So Lee reads about the offer of amnesty and sends an application to Grant in May 1865, but he doesn't include the oath of allegiance that had been recently added by Presidential order. In turn, Grant argues to President Johnson that General Ord, in command of the Department of Virginia, had not promulgated that order in time to reach Lee prior to him sending in the application and thus deserves the general amnesty and a pardon since the lack of the allegiance oath was unintentional.

I'd argue there are a few factors that would suggest Johnson himself then declined this request. First, Lee followed up on the initial request, writing Johnson directly in June 1865 requesting amnesty; it appears he never received a response. Second, we don't seem to have any further correspondence about the Grant-Johnson meeting about Lee, nor did a brief search yield exactly when it took place. If (as is likely) it did take place in June 1865, the timing of it has some significance for a reason unrelated to Lee: it may have very well been the very first real rupture in the Johnson-Grant relationship. Things between the two became so toxic over the following years following that Johnson refused to attend Grant's inauguration and considered Grant an outright traitor.

Last, Lee found out about the additional requirement for an oath of allegiance and swore it out before a notary public on the same day he became president of Washington College - October 2 - and sent it to the President. What happened to that request between 1865 and 1970 is a mystery; Lee assumed it was turned down, and Seward biographer Walter Stahr doesn't mention it, but a good summary of Seward's role in the utter circus of pardoning taking place in 1865 and 1866 (among the many problems with it were the cottage industry getting thousands of applications to Johnson's desk at $5-10 each through an intermediary) reveals a little bit more:

"Seward was deeply involved in the pardon process—both the formalities and the informal conversations. It was Seward, for example, who secured the release from prison and a pardon for Alexander Stephens, the former Confederate vice president and Hampton Roads commissioner. Writing to Fanny in November, Seward told her that she would find their house in Washington to be “the chief resort of the recently rebels.”"

This explains how Seward ended up with the applications to lose and/or give away as souvenirs. That said, despite his pre-war vilification in the South as an abolitionist nightmare, Seward had a track record of trying to work with Southern leaders, even up to almost the outbreak of the war as he fruitlessly attempted shuttle diplomacy to keep Virginia in the Union. He didn't seem to have a personal beef (ala Meigs) with Southern leaders during the war, and by Reconstruction, getting them back into the political arena seems to have fit with his general goal of fully integrating the South back into the Union as soon as possible - one of many reasons Radicals completely broke with him - and it's a bit hard to believe that Seward, the oft-cynical professional politician, wouldn't have viewed a pardon for Lee as helping significantly with that process. Overall, the whole saga including the implied insults of leaving Lee entirely in the dark sounds a lot less like Seward than Johnson, who was among his many other flaws awfully vindictive.

Second, to the general question about the motivation of Ford and Congress, many of the assumptions here seem to look at Lee with a 2022 lens rather than a 1975 one. The historical evaluation of Lee has changed dramatically in the last couple of decades (a review of which probably would make a good top level question) and any attempt at rehabilitation would no doubt be massively polarizing today, but at the time restoring his rights were so relatively uncontroversial that besides the unanimous consent in the Senate and overwhelming House vote on the Byrd resolution, when I went to search through the Ford literature, there's not a single word mentioned about it.

In fact, the most interesting thing to me about Public Law 94-67 is that it performs an act that I'm not entirely sure Congress had even legislated during Reconstruction, as well as being something that has come up in the news quite a bit recently:

"That, in accordance with section 3 of amendment 14 of the United States Constitution, the legal disabilities placed upon General Lee as a result of his service as General of the Army of Northern Virginia are removed, and that General R. E. Lee is posthumously restored to the full rights of citizenship, effective June 13, 1865."

Thus, it is incorrect to call Ford's actions a pardon; they were restoring Lee's rights that had - in theory - been removed by Section 3 of the 14th amendment.

Ford's signing statement is also worth a read.