Why would they bother? His term is nearly over, the executive branch is running smoothly, and he hasn't done anything really crazy. Unless he starts actually being dangerous ("My fellow Americans, we're going to bomb the Soviets today, for real this time!"), I can't see them really invoking the Succession Act...
Really?
Right around this time, just after the midterms of 1986, is when the Iran/Contra scandals began breaking big-time. The stuff we were learning about and the Administration's reaction to it was all pretty scandalizing.
Clearly, one approach to "spin control" might have been to bundle as much of the outrageous stuff together into an alleged out-of-control faction, blame the existence of that faction on the poor Gipper's mental state, and put the President out to pasture. He could conveniently pardon key players just before retiring so they are immune to prosecution, which is sort of what happened OTL--they were blanket-pardoned by Bush.
One reason this did not happen OTL was that the Reagan and Bush administrations wanted to go right on with the same policies; trying to clear the decks like that would have forced them into something of a straightjacket. Which is where I personally think they belonged all along, if not in prison garb, but obviously the American voter of 1988 thought differently.
To my obviously contrarian mind, the real scandals relating to Iran/Contra were firstly that the whole complex of schemes were so successfully kept under wraps before the crucial elections of 1986, not that the outcomes of those elections were a right-wing sweep or anything--probably it would have made little difference to the effective balance of power in Senate and Congress, since that already leaned toward the Democrats--but so much for the power of a free press to keep the nation informed, eh? So much for the checking of ambition by ambition as even a Democratic-controlled legislature did not use its investigative powers to expose this messy flouting of its own legal mandates!
And secondly, that after the deep rot of an administration hell-bent on pursuing a very questionable agenda that had already been rebuffed by the legislative branch by extremely questionable and clearly anti-democratic means, seeking the most shady, seedy, anti-progressive partners they could find around the globe to do it, that the last years of the Reagan administration were
not so wracked with turmoil that both Reagan and Bush were forced to resign or were impeached, not for health reasons but for cause, and Tip O'Neill installed as President according to the line of succession.
The fact that Democratic-controlled investigative committees wound up whitewashing the mess for the President, as well as their failure to derail these schemes much earlier, suggests complicity to me--if not of an openly scandalous kind, than a deep ideological acceptance that the USA's commitment to democratic due process is largely an ideological cover and the real movers and shakers of American policy, regardless of nominal partisan affiliation, do subscribe when push comes to shove to a much more elitist and manipulative view of how things work and should work.
If I seem bitter, this is the very time and very issues where I got good and radicalized; I've had a wry look at American politics and platitudes and professions ever since. This has hardly deterred me from throwing myself in to quite idealistic causes and taking an idealistic approach to not-so-idealistic ones too. I'm a left wing radical
because I am a patriotic American who believes in the rhetoric of the Spirit of '76.
So, given the cynical nature of US leadership on both sides and of the voting public too, apparently, I suppose things were stage-managed OTL about the way we'd expect. Admitting that Reagan was suffering from mental impairment would give the newly-strengthened Democratic caucus in Congress and Senate an excuse to review Reagan-era policy across the board. To my mind of course the Iran/Contra mess was not an excuse but an urgent mandate (in the forceful, not permissive, sense) to do so at once, regardless of the Gipper's state of mind, but apparently not so to Democratic leadership. Had the Dems seized that opportunity had it been handed to them, I don't suppose with my now cynical retrospective view that they'd have substantially changed a lot, but they probably would have upset a lot of cronyish lines of succession and career paths carved out under Reagan's banner in favor of Democratic cronies, and incrementally changed a few things here and there that Republicans wouldn't have liked.
And there was always the possibility that at least some Democrats, and the unpredictably inflammable American public, might blink and take a second look at protruding clues to the mess OTL labeled "Iran/Contra" and start digging, and that the go-along-to-get-along crowd that dominated Democratic policy might get set aside or shoved aside completely by a genuine, probably bipartisan, prairie fire of national outrage and thus upset the whole applecart for all the powers that be.
So the White House and other Republican insiders had little to gain, and possibly (if not certainly) much to lose, by switching Presidents. Letting it ride was their best move. If things did get really ugly for them politically then throwing Reagan to the wolves might have been a good last-minute desperation option, maybe. But his iconic value tended to protect them too if things didn't get too hairy.
Oh yes, this too: I have tangled up two issues here, the Iran/Contra thing and Reagan's health. If things really were going serenely and a happy nation approved the Reagan way of doing things across the board, then the question of whether Reagan should have stepped down should have been a matter of his objective state of mind. But in fact the I/C stuff was writhing around like a suitcase overstuffed with snakes, and Bush the Elder's fingerprints were all over it. Throwing Reagan overboard as a sacrificial distraction would have made no sense as a move to protect his Vice-President, who contrary to his claims was darn well
in "the loop." That kind of spooky stuff was in fact Bush's particular area of expertise, he having been former CIA director for instance, and intimately involved in all kinds of shadowy under the table personal politics in the Middle East and Latin America too. So it would be a gullible, ill-informed American public indeed that would be fooled by tossing the Gipper. Far better, in that sort of dire strait, to toss Bush, because if they were actually forced by American public scrutiny and outrage into actions, the public would not be so ill-informed nor gullible.
I believe that if we are ignorant and easily fooled, it is because at the end of the day, we
want to be. People know that trying to involve themselves in wonky power struggles with things at stake is a social buzzsaw and they are likelier to make enemies than friends. For the most part we let our political types work out their deals and avoid paying attention just as we would if we were sausage-lovers visiting a sausage factory.