Should I cut back this top-heavy aloe-like plant?

If it’s a climbing aloe, sprawling is normal, so a heavy or leaning stem doesn’t automatically mean something is wrong. I’d give it a stake or trellis to lean on and leave the healthy top attached. A support is the least drastic option.

Cutting is an option, but I wouldn’t do it just to make pups appear. You can try rooting a healthy top as a cutting, though success can vary and it may take a while. The original plant may not fill out the bare lower stem the way you’re hoping.

I’d be cautious about burying a long section of stem just to make the plant stand upright. Support it first, and trim only clearly dead leaves. Brown tips and brown pups aren’t necessarily the same issue, so I’d check whether the pups are actually soft or deteriorating before deciding to remove them.