"Why call Piccolo along?"
"To find Gohan a teacher."
What Yinsen had wanted to say was that they should find Gohan a godfather, but he decided against it. Besides, without the threat of the Saiyans this time, he did not know whether Piccolo would still be willing to teach Son Goku's son.
In the original story, the moves Gohan was best at and used most often were all Piccolo's techniques. He even preferred Piccolo's outfit. It was hard to tell who his real father was.
With Son Goku personally teaching him this time, Yinsen wondered whether Piccolo could still hold such an important place in Gohan's heart.
"A teacher?"
"Our strength is too great. Even if we can teach him, we aren't suitable sparring partners for Gohan."
Raditz, whose strength was suitable, simply did not understand cultivation at all. One reason Yinsen brought him along was that Raditz did not want to spend long periods around a group of unfamiliar Earthlings. Although Raditz had little affection for his younger brother Kakarot, they were still biological brothers. Another reason was that Yinsen worried Raditz might secretly cause trouble, so it was better to keep an eye on him.
"That's true, but will Piccolo agree?"
"He will, because he wants to become stronger too!"
No one on Earth understood how to become stronger better than Yinsen and Son Goku. If Piccolo wanted to catch up to them, the best choice was to train alongside Yinsen and Goku.
As for grudges? Piccolo himself probably had not realized that he had no intention of avenging the previous King Piccolo at all. The few times he had spoken of it, he had only said he wanted to defeat Yinsen and Son Goku, not kill them. In other words, the newly born Piccolo subconsciously only wanted