As others said, there are plenty of words to describe the activity itself - copulate, fornicate, etc. What the OP seems to be asking is why there is no polite transitive verb that describes the activity.
My feeling is that using any transitive verb makes the sentence sound offensive or at least impolite. It adds the connotation that sexual intercourse is a one-sided act, not a mutual act.
This has been repeatedly answered. The word you don’t think exists is “know.”
He went home to know his wife.
It fits your required sentence structure, is unambiguous, and it is acceptable usage in polite conversation. (In as much as discussion of sex is acceptable in polite conversation, of course.) And it has been around for a long time.
Another one is “pleasure.” He went home to pleasure his wife. This one is most commonly used for masturbation, in that someone may say they pleasured themself, but it also satisfies your requirement.
“Tupp”
Is this used or does it relate only to animal “mountings”
“Boink” is still in use. You can find it in Janet Evanovich’s novels.
Shall we roll round the haystack with our arms round each other?
While the piggies go “oink, oink,oink”?
We could disco or samba, better yet let’s do la bamba.*
Shall we boink? Shall we boink? Shall we boink?
–Forbidden Broadway’s Parody of “Shall We Dance” from the King & I.
*The king was being portrayed by Lou Diamond Phillips.
The transitive verb “swive” {interestingly, cognate with “swivel”, which was not recorded in verb form until much later} was first recorded in OE, and used in ME by Chaucer, and thus considerably predates the word “fuck”, an import first recorded in the early 16th century. As to why “swive” fell out of fashion, who knows? Sometimes words do. Certainly “fuck” has a better blunt plosive force, something it shares with the other “four letter” words.
Yes but every single example given, scr4, fits within “sounding, imprecise, euphemistic, ambiguous, not in common usage, or are phrases that dance around the subject matter”. I mean really, look at **Ellis Dee’s ** examples: “know” and “pleasure” are both utterly ambiguous, and never used outside of extremely stiff and formal circumstances.
When used in the sentence “He went home to know his wife.”, it’s pretty clear what is meant. One would reasonably assume that he already knows his wife, and this use of the word “know” is in the Biblical sense. One could also argue that “He went home to fuck his wife.”, could be misconstrued to mean that he went home to screw his wife over.
That’s what the OP is looking for. He asked for synonyms for “fuck” “that can be used in polite company or on television.” Coming up with other slang words is a piece of cake.
“Shag.”
“Boff”
Bah! The problem is that people are so fucking uptight about sexuality that they are afraid to use “fuck” in polite conversation. It’s only a matter of time, people, before the President of the United States uses it in his State of the Union address. Sure, that might be decades from now, but it will happen. And some other word will be the taboo one.
Coincidence: just today I was wondering why in English there is either
no names for sexual organs (just “sexual organs”),
or only literary loan words (penis, vagina, testicles),
or plenty of slang names.
Same situation in French and, I suspect, in Spanish and Italian. I don’t know about other languages.
(this is really an IMHO question, but I didn’t want to start another thread)
You can make love without without impregnating, or being impregnated.
Genitalia
Oh, and one more:
Your heart is burst, you have lost your soul.
Even now, now, very now, an old black ram
Is tupping your white ewe. Arise, arise!
William Shakespeare, Othello, 1.1. 87-89
“screw” would come closest to f… in my house - is it not american usage?
It is universal! There is no substitute nor equivalent.
Just as the universal ancient manual gesture of fertility which is understood and offensive in any country around the world.
Very good. 
Thanks, Annie-Xmas.
Noodle.
Woah, I used to live on Coit Road. Heck, I even remember when it was a dirt road.
I went home to **make love to ** my wife.