Undertaker has the advantage here. Thez is not used to these types of matches. Thez might be tough but he doesn't know what the cold steel of a chair feels like.
Unprotected chair shots to the head were illegal in Lou Thesz's day. So no. But that doesn't mean that Lou hasn't taken his fair share of hard shots. He knows what it feels like to be knocked from the ring apron and go flying into the barricade. He knows what it feels like to land hard on the floor outside. He knows what it feels like to be slammed into the turnbuckles or furniture on the outside.
Thez might be able to take a punch to the face. Taker has taken a chair to the head and sat right back up.
Lou suffered broken ribs during match, and still managed to gut it out and win. As Undertaker ever done that? Most modern wrestlers have never done that. Men less tough than Thesz have taken chair shots from the Undertaker and still beat him. If Undertaker brings a weapon into the match, then Thesz will have no problem using it himself if that's what the fans want to see.
Taker has rarely lost by submission despite facing submission machines like Bret Hart.
One of the known instances of Undertaker clearly tapping was due to Brock Lesnar's Kimura lock. A hold that Lou Thesz used frequently. It was one of his favorites. One wrong step and Undertaker would have his arm being wrenched from its socket again.
Undertaker also has mysterious powers that would very likely freak out and confuse Thez. Undertaker has been able to summon a lightning bolt. Thez would have no clue what is going on. If Taker gets caught, he could summon said lightning bolt and probably startle Thez enough to let go of the submission.
Please. Lou did not have time for clown gimmick wrestlers. He took pleasure in roughing them up. He was respected for being a wrestler. Everyone in his era knew this. Honkey Tonk Man was scared to death the first time he locked up with Thesz. Sabu's uncle the Sheik fled and hid under a bus during a match with him. The moment Undertaker does his little undead zombie shtick, Lou will kick the shit out of him. And then probably lock in the Kimura as a "fuck you."
Undertaker will likely wrestle as Big Evil out of respect.
Undertaker likely has learned some MMA counters and could counter what Thez tries. Taker would also use styles that Thez is unfamiliar with.
Lou Thesz was a hooker. Do you understand was that is? He was a catch wrestler. Like in legit MMA. Undertaker may know some maneuvers for show, but Lou knew how to apply them effectively in a real fight. There's nothing that Undertaker would have in his arsenal of subs that Lou hasn't already mastered.
As far as wrestling styles go, Lou knew them all. He could brawl and fly with the best of them from his era. If you have watched his matches you'd know that he threw dropkicks and cross bodies and head scissors all the time.
Soundly? Ha! I have seen modern street fighters that barely leave the ring, let alone become weapon laden bloodbaths. Lou has a winning record in Texas Death matches, meaning that he was brutal and efficient enough to keep opponents down for a 10 count. Lou himself has never been in a cage match, but he helped promote one on a card in Kuwait during the 80's. He knows the finer details. It doesn't take a scientist to figure out how to climb a ladder either.
2/3 falls matches are nothing new to Thesz. Hell even Taker's almighty piledriver is nothing new to Thesz. Lou soaked up 2 of them in a NWA world title match with Bill Longson in 1952.
You can see that here if you go about 7 minutes in...
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...Lou was able to shrug off the affects of 2 piledrivers and still win the 3rd fall.
Undertaker really is in some deep shit here. He's up against an opponent that's not intimidated by him. Lou can keep him grounded on the mat at will and wear down his weak spots at his leisure. Lou's smarter. Lou's faster. Lou has better stamina. Lou wouldn't be afraid to play dirty and bend the rules. Undertaker will likely end up burning himself by the last fall trying to escape all the holds that Lou will have him wrapped up in and will likely end up either being suddenly pinned - sudden pins out of nowhere were a specialty of Lou's - or passing out to the same hold that Lesnar punished him with at Summerslam.
Lou wins.