As you may have guessed by now, considering that nine other answers have already been offered, what you're describing is a common premise in novels of speculative fiction: The story of a young person who has, either consciously or subconsciously, blocked/suppressed/hidden-from-others/whatever his (or her) strong psychic potential, for one reason or another -- such as the fear of being burnt at the stake as a witch -- but who ultimately ends up learning to make good use of telepathy and/or other "psychic powers." But there was one old science fiction novel in particular which sprang into my mind first when I was reading your post, and I see it hasn't been mentioned in any of the previous answers you've received, so I'm going to take the plunge! (If this doesn't work, I may have to reexamine some of the candidates which popped into my head a bit later on.)
Pstalemate, by Lester del Rey. First published by G.P. Putnam's Sons in 1971. (Which matches nicely with your recollection of running across this story in the early 1970s, when it would have been a new arrival in bookstores and libraries.) I checked out a copy from a library, somewhere around the late 1980s or early 1990s.
Here's a cover-scan from what was apparently the first edition. It might ring a bell in your memory? (There were other editions later, with various other cover illustrations, but I'm not going to cram them all into this post.)
Partially from memory, and partially from online resources to assist my memory, I will give a quick summary of the plot -- mainly how it is set up in the first few chapters -- to illustrate how this story matches pretty well with the points you listed.
I don't recall any exact dates being given, but the book seems to be set in what was "modern times" when it was written. In other words, somewhere in the USA, with appropriate technology and social customs from around the late 60s/early 70s. (Therefore, no "futuristic" technology was in common use that didn't yet exist in reality, such as flying automobiles, or interstellar spaceships, or even everybody having a personal smartphone in his or her pocket.) That matches with your recollection of what would have been an "ordinary, everyday setting" at the time you read it.
The protagonist is Harry Bronson, a sincere young mechanical engineer. Early in the book, he is persuaded to let an old friend at a relaxed social gathering test him for telepathic ability. Nothing fancy; the friend (Dr. Lawson) will just go through a deck of cards, one at a time, holding each one so that he and at least one other witness can see what the card is, but Harry can't, and asking Harry to name whatever card he thinks is currently showing. They keep written notes. On the first run-through, Harry says whatever guess pops into his head for each card, and after they've gone through an entire deck, he is told his score was a big fat zero -- he didn't get a single one right!
Harry is not heartbroken by that result. He humorously suggests that this means he is one of the least psychic people alive, or words to that effect. Dr. Lawson argues the point, claiming that total failure is just as significant as a perfect success rate would have been. If Harry were making a long series of purely random guesses, he should have gotten at least a couple of 'em right by the law of averages -- and the fact that he didn't is strong evidence that Harry has considerable psychic potential, and something in his brain is subconsciously refusing to let him make a correct "guess" about any of those cards, for fear of what might happen! (In other words, if a correct "guess" started to creep into his head on a psychic wavelength, his subconscious supposedly stifled that thought in a hurry and substituted a guaranteed-to-be-wrong "guess" instead, which would be what Harry ended up voicing at the time.)
Harry finds that argument less than totally persuasive (and I'm not saying I blame him), but Dr. Lawson hypnotizes him to try to overcome the alleged subconscious block, and then they test Harry again, on a double deck of playing cards (apparently including 2 Jokers per deck, for a grand total of 108 cards in a row, according to one reviewer).
This time, Harry finds himself visualizing a card each time, and writes down what he "sees" in his mind's eye, and when he's done, he is told that he got a perfect score --- 108 matches out of a possible 108, which is so incredibly unlikely to happen by sheer chance that it's not even worth considering the possibility that nothing more than a statistical anomaly was happening here. (Again, all this is according to Dr. Lawson, who seems to have a secret personal agenda here, as I noticed when I was first reading the book.) Harry finds himself feeling terrified by this discovery and the implications, although he is not sure why.
As he tries to go about his everyday life, he keeps experiencing more and more evidence that he is now an "unblocked" functioning telepath whose brain is learning how to do things it never did before (not that he can recall, anyway). He touches base with a girl named Ellen whom he hadn't seen in a long time. It gradually becomes clear that Ellen is also telepathic, and that other telepaths exist in the world (including Harry's mother and also one of Ellen's parents -- each of these two young people apparently had inherited the talent), but you don't run into functioning telepaths very often because full-fledged telepathy has this nasty habit of driving the possessor totally insane. Harry's long-lost mother, for instance, is still locked up in an asylum, and has been for many years, ever since she set the house on fire when Harry was a ten-year-old boy, nearly killing him. Other telepaths either go totally psychotic or commit suicide or otherwise are removed from general circulation. From online reviews, I get the impression this usually happens by around age 30 or thereabouts. (I can't remember if an average age was stated in the book.)
Just in case Harry didn't have enough to worry about yet, the author allows him to start noticing he also has a precognitive power which gives him glimpses of himself having psychotic hallucinations or something along those lines -- about three months in the future. This gives him a hard-and-fast deadline for trying to figure out a way to avoid the descent into madness which his mother and various other telepaths have suffered. (Ellen isn't feeling all that mentally stable, either. Meanwhile, the two of them appear to be falling in love.)
I don't fully remember the details of the novel's climax (although some online resources helped remind me), but there does seem to be a happy ending for Harry and Ellen. I won't ruin such details as I remembered and/or have found by Googling; I'll just say that while I found much of this book to be very depressing when I read it, many years ago, I found some comfort in the thought that the protagonist and his girlfriend were apparently going to "break the cycle" by staying sane for many years to come while raising children. (They either had married, or were planning to marry, by the end of the novel.)
If that description, or any substantial portion of it, stirs up some old memories in your head, then we're probably talking about the same novel. In case you decide you want to read or reread it to check, I did a little Googling. Pstalemate does not appear to have had any English-language reprint editions since the 1980s, and a little poking around on Amazon indicates that no one has bothered to put together an official Kindle edition yet, but the good news is that if you want to read a second-hand copy of this and see if it matches up with your vague memories from the 1970s, it won't cost you very much to satisfy your curiosity.
Amazon currently has used mass-market paperback copies available (mostly from other vendors) for as little as USD $1.99 plus shipping & handling, or USD $5.89 (and up) if you want the shipping & handling costs to be included in the basic price you pay. (Either way, there are likely to be additional sales taxes, but I don't know if yours would be the same as what shows up on my own screen when I look at the options, so I won't try to predict them for you. I also don't know if you currently reside in the USA -- if not, the shipping costs might go up, or you might want to order from a different version of Amazon, or some other book-selling website based in another part of the world, instead of using Amazon.com?)