Morphy vs. Steinitz
Steinitz didn't do anything that Morphy would not have understood, and Morphy's tactical superiority would have given him a decisive advantage. But Steinitz explained himself in detail, and so his gifts to succeeding generations of players far exceeds that of Morphy.
Anderssen, too, understood what Steinitz was beginning to do, though it was too late in his lifetime to take full advantage. Morphy, from his own commentary, viewed Anderssen as the only opponent of sufficient skill to truly test the soundness of an opening (as he did with the Evans Gambit in the opening game of their match). Steinitz used Anderssen in much the same way when testing his positional theories - it doesn't count for much unless the opponent can really make you pay for the imperfections. The strength of raw tactics had to be more or less tapped out before folks could realize there was more to the game.
Anderssen died in 1879. I know of no post-1860 publication of Morphy commentary involving testing the soundness of openings.

Dsmith42
I agree with much of what you say, though I would like to point something out: The strength of Morphy's game was in his position abilities, not his tactical chops. The reason the saying "Morphy's pieces do not retreat" came about was because he grabbed squares when they should be grabbed; the attacks flowed naturally after that. The sacrifices he was known for are what Alekhine called the "trinkets and baubles" of Morphy's play. And you shouldn't sell Steinitz short on his tactical abilities, either; look at the times he clobbered Tchigorin, Tarrasch, Lasker, von Bardeleban--the man had some serious tactical chops, too. To make a gross comparison, Tal and Petrosian both had incredible tactical vision, though you wouldn't think that from their styles. So, to put my two cents in on what a match between Steinitz and Morphy would come down to is this: Morphy was raised in an environment where his gift was recognized early on and put on display on Sundays, so he was raised as a performer rather than a chess player as we now know them. Steinitz put his heart and soul into the game and always had to battle Morphy's ghost, so he had something to prove to the world--maybe even to himself. In judging Morphy's ability to adjust his game (ala the match with Harrwitz) versus Steinitz's stubberness in repeating lines, as he did throughout his career, to prove he was right, I feel that that weighs strongly in Morphy's favor. Each would have scored impressive wins against the other, but when faced with a new level of play I think Morphy had the ability to raise his game to it--I don't think Steinitz's game had the extra level to go to.
... In judging Morphy's ability to adjust his game (ala the match with Harrwitz) ...
Do we have the means to judge the difficulty of adjusting to an 1858-Harrwitz compared to the difficulty of adjusting to an 1886-Steinitz?

... In judging Morphy's ability to adjust his game (ala the match with Harrwitz) ...
Do we have the means to judge the difficulty of adjusting to an 1858-Harrwitz compared to the difficulty of adjusting to an 1886-Steinitz?
I don't know if we have the means to make that comparison, but what I'm saying is look what Morphy accomplished virtually without effort versus what Steinitz labored hard for. Hard work can defeat pure talent but I think Steinitz was at the limit of his talent while Morphy still had more in reserve. Talent plus hard work produces monster chess players, and I think that is what would have happened if Morphy had been tested.
... look what Morphy accomplished virtually without effort versus what Steinitz labored hard for. ...
Did Morphy have to face the same sort of chess world that Steinitz faced around 1886? How many international tournaments had there been by the end of 1858?

You're talking about competition while I'm talking about strength. With all the advantages of increases in theory, with all the extra games to study, with all the tournaments, Steinitz showed no marked increase in strength over Morphy. And Morphy achieved his strength WITHOUT the benefit of the competition. That's the point you seem to miss, I don't care how many tournaments Steinitz played in or who his opponents were, it didn't seem to help to get him to a place stronger than Morphy probably because he was so stubborn: stubborn as a youth, stubborn as a champ, and stubborn as an ex-champ. Morphy was not as closed minded (no pun intended), he would have showed Steinitz what Steinitz and the rest of the world had yet to discover. Face it, even with all the advantages that Steinitz enjoyed in a world after Morphy, he never showed a clear superiority in understanding over Morphy, and for that, Morphy is the better player.
... With all the advantages of increases in theory, with all the extra games to study, with all the tournaments, Steinitz showed no marked increase in strength over Morphy. ...
How could he? Was Steinitz playing against 1858 opponents?
... I don't care ... who [the opponents of Steinitz] were, ...
Can one reliably assess strength without thinking about the strength of the opponents?
... even with all the advantages that Steinitz enjoyed in a world after Morphy, ...
Do we have reason to rule out the facing of 1880s opponents as a substantial disadvantage?

Didn't Steinitz lose to Maurian (Morphy's mate) who himself got rekt by Morphy (w/ knight odds) for at least some time.
Either way I don't doubt Morphy had more than the requisite ability to beat Steinitz. Although it would've been difficult against prime Steinitz especially considering Morphy hadn't played a single competitive game for years.
Didn't Steinitz lose to Maurian (Morphy's mate) who himself got rekt by Morphy (w/ knight odds) for at least some time. ...
Didn't Morphy lose a game where Barnes replied to 1 e4 with 1... f6 ?
http://www.chessgames.com/perl/chessgame?gid=1266586

Except Barnes was one of the strongest players at the time. Maurian wasn't even a chess master. Only played Morphy and managed a win/draw against Steinitz.
Try harder breh.
What matches/tournaments did Barnes win? After 1 e4 f6, do you suppose that Morphy made a mistake somewhere?

i put steinitz as slighly superior to morphy, but morphy as more naturally talented. steinitz simply grew with the most stimulating environment available to maximize his potential.
i put both as significantly inferior to lasker though who even in old age gave capa a hard time.
here is a brilliant dissection by im silman on how steinitz changes the chess world and how great he is as a player.
https://www.chess.com/article/view/steinitz-changes-the-chess-world