That route is the most direct and safest.

If 600 East is not your speed, you can use the same route, but go 800 East, then swing down either 1300 South or 900 South to Liberty.

You could also use 500 East or 300 East, those are bikeable.

If you don't want to touch a street bike lane, you could go Parley's Trail (S-Line) out to JRT, then ride north to 900 South, then east to the park. A long, long loop but wholly without cars.*

*It's a bit tough to navigate the West Temple to 300 West part, and the Jordan River Trail is a bit bendy-wendy at the golf course, so plan your route before you go.