What exactly could the NLRB have done to make the change you're referencing. At least some of the election process standards are set forth in statute. Last I checked, administrative agencies can't override statutory enactments. How could they have done so on this issue?
If it were so easy, why didn't the NLRB do so after the unpopular EFCA failed in Congress?
What exactly did Obama fail to do on this issue? Even Democrats balked at the EFCA.
What specific promise did he break on this issue? What was put on his desk for signature that he refused to sign despite his previous promise?
You misunderstand the labor law system.
The sitting President appoints a majority of the NLRB.
The NLRB then sets (or re-sets) all kinds of rules - some of which require regular rule-making, some which don't. Depends on the specific issue, the specific statutory language, and even prior Court rulings. Some are set by the formal rule-making you alluded to, and maybe followed by Federal Court of Appeal review (under the "deferential" analysis they give agency actions). But MOST of their rules are set by Board/General Counsel pronouncements of policy (i.e. how they interpret certain statutory phrasing or prior regulations), and/or specific Administrative Law Judge rulings - some (not all) of which are also tested by Circuit Courts of Appeal under the "deferential" analysis they give agency actions - but still NOT by the formal rule making process you alluded to.
Thus, for example, during the Obama Administration, rules about the use of/meaning of/impact of things like employment handbook policies - on topics like at-will employment, social media use, etc. - were changed with no new "regulations" or "statutory amendments." (Statements of, for example, at-will employment, that were understood to be legal for decades, and previously declared legal by courts, were suddenly declared illegal based on the possibility that extreme interpretations offered by no one maybe potentially someday might harm employee rights. Same process in other areas - the Board changed how they interpreted the meaning of legal phrases like "bargaining unit" and "community of interests" to allow a union to represent "micro" units, and not require them to show "majority" support to represent larger groups (more on that term below).
No amended statutes or regs were offered or seen to be necessary. Nothing "placed on the President's desk." They just changed the rule - like Obama SAID he'd do with "card check."
The rules set by the NLRB include how/when to hold secret ballot elections.
The statutory language "requires" a "secret ballot" election when there is a "showing of interest" at the 30% level.
The statutory purpose of the election is to determine whether a "majority" of the relevant employees favor unionization. (Ties go to the employer. The union must get 50% plus 1 of the votes cast, minimum.)
The unions have always argued that if their "showing of interest" was already more than a majority, no election should be held or is required by statute. "A majority is a majority."
They want to gain the right to represent employees solely through "card check" because once the unionization effort is discovered, unions historically lose support between that date and the election date. (The parties disagree on why. Unions say the companies violate the law and illegally threaten/retaliate against pro-union employees. Companies say the employees have typically been lied to, bothered and bullied, and vote NO in secret ballot elections.)
The Obama NLRB could have simply adopted that rule, taking the position that the NLRA requires an election when 30% employee support has been shown, but that "50% + 1" entitles the union to represent the employees - and where the unions shows "50% plus 1" through card check, no election is proper.
Instead, they "elected" to "use" the "EFCA" to "amend" the statute. Planned failure in advance. Once the unions saw EFCA coming, instead of just a policy change, the ones with experience knew. Bubba at the Hall still believed. But the guys at the top knew - "card check" was just another lie for votes.
Remember those angry white union guys you Clintonistas say cost her the election?
They have long, quiet memories.