Well, it would indeed be burdensome if it was taken in a way not intended by the words, namely, literally. Jesus uses hyperbole a lot in His teaching, so that it would be very memorable to his disciples, who were chosen to convey His teaching to the nations—and even to their audience.
The meaning of 'cut off rather than' is a hyperbolic way of teaching us to avoid occasions of sin. If [x] causes you to sin, and sin leads to death, avoid x. More than anything else, it's common sense crystalized in a hyperbolic littel packet which is easy to remember and hard to forget.
The question assumes that Christians have no more help for overcoming sin than unbelievers, which is not true: they have the Spirit of God in them. And "if by the Spirit you put to death the deeds of the flesh, you shall live." (Rom 8:13)—St. Paul's version of the same teaching.
John 15:5 (DRB)
I am the vine; you the branches: he that abideth in me, and I in him, the same beareth much fruit: for without me you can do nothing.
Something could also be said of the difference between the Law of the letter, and the Law of grace, the latter of which doesn't change the moral requirements, but involves as a constituional part of the Covenant, the grace won for us on the Cross, of which we are made beneficiaries when we confess to God our sins: that is, mercy.
John 1:17(DRB) For the law was given by Moses; grace and truth came by Jesus Christ.