Gravel v. U.S. was not, strictly speaking, about Gravel's right to read the PP into the Congressional Record, even though his immunity from prosecution from doing so was upheld as a part of the decision. (It was, for all intents and purposes, merely reasserted. This was not the main subject of dispute.) This case was more about whether the immunity he enjoyed extended to his aide who had helped him do this and whether his releasing the papers to a private publisher was also covered under the Speach and Debate clause.
The Court ruled the aide was covered within the legislative sphere but that the private publication was not. In addition, the Court ruled against Gravel on the matter of whether his aide could be compelled to testify in an investigation of any wrong-doing in having the report privately published.
Also pertinent to the decision was the fact that the report was put into Gravel's possession via an intermediary from Ellsberg, he being the one who had leaked the report to the New York Times. The ruling implied without ruling directly since it was not relevant, and Gravel's attorney's accepted, that had he personally stolen the report or acquired it illegally through an act he initiated, the Speech and Debate clause would not have applied, and both he and his aides could have been prosecuted.
Gravel's involvement came in the amidst the efforts of the Nixon administration attempting to stop the New York Times from publication of the report leaked to them by Daniel Ellsberg. Ellsberg had been attempting to get Senators to *publish* the report on the premise that the Speech and Debate clause would protect them, but all had refused. He then offered the report to the Times, which went ahead with publication and initiated the NYTCo v the U.S. case that ended up ruling in favor of the newspaper.
The Gravel decision has been noteworthy since in setting the limits of executive, legislative and judicial immunity, sometimes against those claiming it.
Anyway ... your point is certainly valid in the sense that if members of Congress have access to documents they could rightly read them into the Congressional record without fear of prosecution. And clearly Gravel is to be commended for what he did. The key in the present circumstance is whether they have legal access to those documents or had been given those documents by some other party. They can't, IOW, just go to the NSA or CIA and say, "Give me all your documents," because those organizations won't give them, generally speaking, nor could they break in and steal them.
Gravel v. U.S., complete decision with footnotes