Mount Myōboku's oil really did lose its efficacy soon after leaving its place of origin. Its "shelf life" was rather short, for a simple reason: the oil was rich in Natural Energy, and once it reached an area where Natural Energy was thinner, it naturally faced the problem of "evaporating."
However, this loss of efficacy only occurred when the oil was stored normally, without any special treatment. The fact that Mount Myōboku's oil was hard to preserve didn't mean it couldn't be preserved... Sealing Jutsu could extend its shelf life, for one.
There was even a simpler, "physical" solution: make a large pouch out of toad skin to store the oil... Toads on Mount Myōboku carried large quantities of oil inside their bodies.
Jiraiya chose to turn Yuzuru down partly because he didn't want Mount Myōboku's things to leak out. As Mount Myōboku's successor, he could use the oil freely, but handing it over to a third party didn't feel quite right.
Another reason was that the oil was extremely dangerous to use. It could draw in Natural Energy, and one careless mistake could turn the user to stone.
And then there was the fact that giving Mount Myōboku's things to someone from Ryūchi Cave just felt strange. The Three Great Sage Regions might be described as standing side by side, but in reality, calling them "opposed" wasn't wrong either.
"If you're trying to work toward the same goal as me, then there's no need to seek out Mount Myōboku's methods... Mount Myōboku's methods are different from Ryūchi Cave's. Would either side really have anything useful to learn from the other?" Jiraiya said, shaking his head.
He thought Yuzuru was learning Senjutsu, but in reality, Yuzuru wasn't thinking that far ahead. It was impossible to say he wasn't interested in