Let E be an elliptic curve over $\mathbb{Q}$. Is finding a point of infinite order on E as hard as finding a generator for it?

Let E be an elliptic curve over $Q$ with positive rank $r$. I am looking for algorithms which find a rational point on $E$. I think the algorithms find points with the lowest height. But when I use Magma to find the generators of the elliptic curve $y^2 = x^3 - 1563056672958141*x$ (which has rank 2, with generators P1=[48408867,194361588954] and P2=[48432972,194700535386]), Magma returns $Q=[260509445493025,-4204701905638250451710]$. I know $\hat{h}(Q)>\hat{h}(P1),\hat{h}(P2)$!My question is: finding a rational point on E is easier than finding its generators?

Let E be an elliptic curve over Q. Is finding a point of infinite order on E as hard as finding a generator for it?

Let E be an elliptic curve over $\mathbb{Q}$. Is finding a point of infinite order on E as hard as finding a generator for it?