Q: How does this leave any room for Congress to tell states the manner in which electors are certified?
Q: Isn't the Constitution pretty clearly saying the state legislatures get to decide such things?
The state legislatures do not certify electors. Only the vote is certified and only by the electors appointed under state law.
The Electors shall meet in their respective states and vote by ballot for President and Vice-President, one of whom, at least, shall not be an inhabitant of the same state with themselves; they shall name in their ballots the person voted for as President, and in distinct ballots the person voted for as Vice-President, and they shall make distinct lists of all persons voted for as President, and of all persons voted for as Vice-President, and of the number of votes for each, which lists they shall sign and certify, and transmit sealed to the seat of the government of the United States, directed to the President of the Senate; ... [Twelfth Amendment Election of President] (emboldening added)
To that end, the proposed ECRA (Summary), merely reflects those requirements. With respect to the transmission of the sealed certificates, the state still retains authority for selecting the individual responsible for doing so; but requires only that the selection cannot be changed after the election.
The bill specifies that the choice of electors must occur in accordance with the laws of the state enacted prior to election day.
Additionally, the bill identifies each state's governor (unless otherwise identified in the laws or constitution of a state in effect on election day) as responsible for submitting the certificate of ascertainment identifying the state's electors.
Q: After all, isn't it up to Congress (as a whole) to tally the votes?
Q: And doesn't each Congress get to make its own rules?
Q: What weight would a prior statute carry in this regard, and how would that be different from a rule change?
Each House may determine the Rules of its Proceedings, ... [Article I, Section 5, Clause 2]
The Constitution’s Rulemaking Clause authorizes the House of Representatives and Senate to establish rules by which each will conduct its own business.
When Congress meets for the purpose of selecting the president and vice-president, it is not "conduct[ing] its own business" — it is conducting the nation's business. This can only be accomplished under the Constitution or a suitable law and only under a Concurrent Resolution authorizing the two houses to work together. (See, for example, S.Con.Res.1 - A concurrent resolution to provide for the counting on January 6, 2021, of the electoral votes for President and Vice President of the United States.)
Q: [D]oesn't this mean that the mechanics of the counting process is left to however Congress decides to do it—according to whatever rules (or lack thereof) it wishes to proceed with at that time?
Q: How can today's Congress bind a future Congress with a statute in any way?
See above, with respect to the nation's business.
Q: [T]he Constitution also says "the person having the greatest number of votes shall be the president". Isn't this clause pretty... crystal clear?
Q: What part of this leaves any room for doubt about who becomes president, let alone whether the VP would have any say in the matter, to warrant the ECRA?
The more complete text allows for the possibility that no person has a majority for selection as president.
... The person having the greatest number of votes for President, shall be the President, if such number be a majority of the whole number of Electors appointed; and if no person have such majority, then from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President. ... [Twelfth Amendment Election of President]
In the case where no person has a majority, the House selects the person who shall be president.
Legal weight of the Electoral Count Reform Act (ECRA)
The general authority for Congress to make law is given in Article I, Section 8, Clause 18.
To make all Laws which shall be necessary and proper for carrying into Execution the foregoing Powers, and all other Powers vested by this Constitution in the Government of the United States, or in any Department or Officer thereof.
Because the Twelfth Amendment delegates to the House and Senate the power to choose, under some circumstances, the president and vice-president, respectively; it follows that, should Congress find it necessary and proper, a law may be written to address any problems that may arise during or after a presidential election.
The Twelfth Amendment, as written, anticipates that only one list of votes will be submitted by each state for each office. However, after the Civil War, there were, from some states, two or more lists of votes claiming to be valid. This (and other reasons) led Congress to write the Electoral Count Act of 1887 (ECA). It was necessary and proper because the Twelfth Amendment didn't anticipate those problems.
During the 2020 Presidential elections, additional problems arose which suggested problems with the ECA. Some members of Congress agreed that changes were necessary and proper so legislation was introduced, the ECRA, to resolve those problems.
If the proposed changes prevent the problems from arising again, then future presidential elections will occur as the Twelfth Amendment anticipated.
Furthermore, the proposed legislation (S.4573) includes the Presidential Transition Improvement Act, as necessary and proper, to address problems that arose during the transition period after the 2020 Presidential Election.