I've always seen Varys as a pragmatist. He let Ned die instead of saving him because he understands you cannot always do what is "right." So why didn't he backtrack after Tyrion told him he was loyal to Dany?
At that time, Tyrion hadn't decided which camp he belonged in and Varys was recruiting. Support of the Hand of the Queen would have been crucial in the designs he had. After all if he'd succeeded in removing Daenerys by assassination, the power would have fallen intp Tyrion's hands as Hand of the King, Protector of the Realm and Lord Regent during the interregnum.
Varys was still hopeful by the end of that exchange that Tyrion might agree to join him as evident by his parting note.
VARYS: I've served tyrants most of my life. They all talk about destiny.
TYRION: She's a girl who walked into a fire with three stones and walked out with three dragons. How could she not believe in destiny?
VARYS: Perhaps that's the problem. Her life has convinced her that she was sent here to save us all.
TYRION: And how do you know she wasn't?
[Varys seems he doesn't have any answer before he changes the tack]
VARYS: Then there's the problem of Jon Snow. Perhaps it's actually a solution. You know them both. Tell me, who do you think would make a better ruler?
TYRION: He doesn't want the throne. That's why he bent the knee.
VARYS: Have you considered the best ruler might be someone who doesn't want to rule?
TYRION: We're discussing treason.
VARYS: Don't pretend you haven't thought about it.
TYRION: Of course I've thought about it. Thoughts aren't treason.
VARYS: He's temperate and measured. He's a man, which makes him more appealing to the lords of Westeros, whose support we are going to need.
TYRION: Joffrey was a man. I don't think a cock is a true qualification, as I'm sure you'd agree.
VARYS: And he's the heir to the throne. Yes, because he's a man. Cocks are important, I'm afraid.
[Now Tyrion has no answer so he changes the tack]
TYRION: What about my earlier proposal? They could rule together as king and queen.
VARYS: She's too strong for him. She'd bend him to her will, as she already has.
TYRION: He could temper her worst impulses.
VARYS: As you have? You're drinking quite a lot. A Targaryen father and a Stark mother. Jon's the one man alive who might actually be able to keep the North in the Seven Kingdoms.
TYRION: How many kings and queens have you served? Five? Six? I've lost count.
VARYS: You've always known my reasons.
TYRION: At a certain point, you choose a person you believe in, and you fight for that person.
VARYS: Even if you know it's a mistake?
TYRION: I believe in our queen. She'll make the right choice. With the help of her loyal advisors.
VARYS: You know where my loyalty stands. You know I will never betray the realm.
TYRION: What is the realm? A vast continent, home to millions of people, most of whom don't care who sits on the Iron Throne.
VARYS: Millions of people, many of whom will die if the wrong person sits on that throne. We don't know their names, but they're just as real as you and I. They deserve to live. They deserve food for their children. I will act in their interest, no matter the personal cost.
TYRION: So, what happens to her?
[Varys looks at Tyrion in a way that implies that Tyrion knows the answer - She'll die]
TYRION: Please. Don't.
VARYS: I've spoken as honestly as I can. Each of us has a choice to make. I pray we choose wisely.
S08 E04 - The Last of the Starks
Notice this conversation. Varys cautiously begins by talking about Tyrants to judge which side does Tyrion stand on. After that he tries to reason that Daenerys' troubled life might be the reason she might become a tyrant in her zeal to save people who don't need saving. Tyrion proves his loyalty by asking him how does he know that Daenerys really isn't meant to save everyone? Varys has no answer so he tries to come at it from a different direction.
Now he starts talking about Jon Snow being the possible solution. Tyrion reluctantly and implicitly admits that Jon might be better but he doesn't want the throne. Varys becomes more confident by this small victory, and tries to push the point. Tyrion warns him that what he's saying is treason. Varys, still hopeful that he was making progress in turning Tyrion to his side, keeps pushing. To further reinforce his point, he enumerates Jon's virtues and points out that he's after all the heir to throne as both Rhaegar's and Daenerys' heir.
Tyrion seems troubled so he asks Varys one more time about binding the two Targaryens in bond of marriage so that they may both share the throne. Varys rejects his proposal once again by claiming that Jon's too weak for Daenerys and she's already bent to him to her will. Even if they wed Jon to Daenerys, it will still be Daenerys who would rule. Tyrion however hoped that he might be able to control her, Varys counters that Tyrion himself couldn't control her. Now he seems to plead to Tyrion by talking about integrity of the realm and saving the union of seven Kingdoms.
Tyrion gets visibly agitated and annoyed at Varys' persistence since it is clear to him now that Varys means to do something and it is not just idle talk the way it began. He accuses him of being a habitual turncloak who's unable to keep faith in anyone. Varys then implores the humanitarian side and says that millions of people will die if the wrong person sits on the throne.
Tyrion weighs it in his mind, seemingly he seems to be in agreement about removing the Queen from the throne. But in what manner? That's what he asks Varys. Varys implies she'd die. At that Tyrion explicitly tells Varys that he won't stand by it. Abdication? Maybe. A coup? Perhaps. But assassination? Tyrion won't support her murder.
Varys however is still hopeful that he may have in fact swayed Tyrion. After all murder is just one step away from forcing a monarch to abdicate. If Tyrion seemed okay with the coup, he'd accept murder as fait accompli. So before leaving, Varys tells Tyrion that they both have a choice to make and he hopes that Tyrion chooses wisely. No doubt in his mind Varys thought Tyrion would make the choice to stand with him or at least he'll choose not to interfere. But Tyrion had made up his mind right after Varys implied murder. Still, Tyrion did not go to Daenerys just then. He wanted Varys to repent his treason quietly but then he saw Varys speaking to Jon alone suspiciously. Tyrion knew now he had no choice but to go to Daenerys. It was better for him personally if Daenerys heard about the plot from him rather than Jon Snow.
His intentions were already clear so there wasn’t much point in trying to cover it up now. Plus the two are friends D&D call Varys Tyrion’s best friend saying something like the following in the Inside the Episode for “The Last of the Stakrs”.
For Tyrion he’s really saying goodbye to his best friend.
They’re friends and Varys is trying to help his friend and doesn’t really feel like lying to him. On top of this he was probably also trying to rally support away from Dany to Jon’s side so he has less opposition.