Why do you believe that NOBODY would find out... when he is a 'public figure'?
If he was just a 'regular joe'.... I would agree with you alot more.
But when you appear on tv every single day....random strangers will 'ask for your autograph'... or ask to 'take a photo' with you... or just hang around you at a bar & strike up a conversation... pretty much everywhere you go.
Most folks with his amount of money & recognition use 'fake names' when they check into hotels.
Somebody will spot you.
And snap your photo when you are not paying attention.
Then post it on social media.
Ssshheeiiiit... some folks will take your picture... just so they can sell the pics to a tabloid, or TMZ.
Look at the line of fans standing right outside the Rockefeller studios every single morning... for a NEWSCAST.

A whole 'new crowd' of random folks be out there... every single day. 365. Rain or Shine.
While MILLIONS more are watching at home. Or at work.
These are the same folks who might approach him / or instantly recognize him on a crowded NYC street.
Tourists. Soccer Moms. Regular Joes. Etc.
They would pass by and make a comment.... or ask for a photo / or a quick autograph.
Hell, they probably even know what his wife looks like.
When you are a public figure... 'being discreet'..... (especially in order to cheat on your wife

)... is much easier 'said than done'.
Now I completely agree.. they wouldn't know what he does 'behind closed doors'... but he gotta get to a destination FIRST... in order to actually 'close those doors'... so the only place he could take the women is to a local hotel somewhere nearby. (where he would get spotted.

)
So all it takes (to ruin his plans) is some random tourist in a hotel lobby... begging for a photo OP.

Or some waiter/waitress (with a cameraphone) who can't wait to tell their friends... 'guess who came to my restaurant today'.

Or some (minimum-wage) hotel worker... looking for a quick TMZ payday.

Or a chatty doorman of a luxury building... who just can't keep a secret.

Or some nosey Uber driver... with a dashcam & a big mouth.