In PA, IIRC, the legislators sued to overturn the judicial rewrite of the mail-in ballot law. The state supreme court, which authorized that rewrite dismissed the suit stating that the legislature hadn’t sued in time (it was after the election), despite the judicial rewrite occurring only a short time before the election.
The legislatures wrote the rules, the executive and judicial branches changed them, and the legislature was denied relief to reimpose the original rules because they “didn’t react fast enough”.
In GA, there were not one but apparently two separate “secret” consent decrees between the executive branch and the democrat party. Neither was publically announced until shortly before the election, IIRC. Neither had as a party the legislature, so the legislature was not informed until too late.
In both PA and GA, as well as a couple of other states, the legislatures voted alternate slates of electors. in MI, which did the same, the executive branch tried to intimidate and prevent those electors from voting their alternate slate.
I am unclear whether a state legislature can sue its executive and/or judicial state branches in federal court, much less at SCOTUS. SCOTUS has original jurisdiction in disputes between states, but has (rightfully, IMHO) refused to participate in most intra-state squabbles.
The sad conclusion, to me, is that the legislatures set up their rules, which were unconstitutionally changed just before the election, and nobody seems to have standing and/or jurisdiction to adjudicate the wrongs. The legislature could possibly simply order the seating of alternate electors, but the administration of these elections were assigned to the executive branch, trusting that they would follow the law as written.
The cheat, even when discovered, in this scenario cannot be resolved by peaceful means within the state governments. The side that cheated, in most cases, controls the apparatus appointed to ensure proper administration of the legislature-defined rules for the election.
What specific actions would you have those legislatures take?
The legislatures should have convened and voted for their own separate slate of electors.
This was NOT done yesterday. The reports you saw about alternate slates of electors being assembled was done by the electors themselves and/or the state GOP organizations, not the legislatures.