Levelling without moving the date
Levelling shifts work into float until nobody is committed twice. What it costs you is the float, and that is a real trade.
There is no single correct levelling
Levelling is a heuristic, not a solution. Considering activities least-float-first gives one answer; earliest-start-first gives another; longest-duration-first gives a third. All are defensible and they produce different schedules.
This is why any levelling result has to state what it optimised for. A tool that shows a tidy histogram without saying which objective produced it has hidden the only part that was a judgement.
Float is a shared resource
An activity moved to resolve an overload has spent float it can no longer use for anything else. The finish date is unchanged and the plan is more fragile, because that activity now has less room before it starts pushing the end.
A perfectly level histogram with everything pushed to its late start is a plan in which nothing can slip at all. That is usually a worse plan than one with a mild overload and some slack left.
When levelling cannot fix it
Sometimes the float simply is not there. There are exactly four honest responses: add capacity, cut scope, accept a later finish, or accept the overload and its consequences.
Each has costs that arithmetic cannot weigh, so a tool should present the finding rather than pick. Silently extending the project to tidy the chart answers a question nobody asked.
The flat-spread simplification
Effort is normally spread evenly across an activity's duration, because the alternative needs a profile nobody has. Real work is front-loaded, back-loaded or lumpy, so a level-looking chart can still hide a genuine peak.
What this does catch reliably is the structural problem — one person committed to two activities across the same days. That is an arithmetic fact about the plan rather than a guess about someone's week, and it is the version worth acting on.
Shared activities and pooled capacity
Two people on one activity should be modelled as sharing it, not as each doing all of it — otherwise a plan silently requires twice the people it lists.
A capacity above one means interchangeable people under a single name. That is fine for a pool of similar skills and wrong for two specialists who cannot cover each other; model those separately or the plan will look feasible when it is not.