If you think his speech projected unity then you are very much not engaged with the larger political climate out there.
Biden's speech may have played lip service to unity, but it went out of its way to unnecessarily touch third rails that were obviously going to be divisive, and that word divisive.
Conservative outlets had a field day with the content he put out there in his speech, again unnecessarily, as they used it to show how he was taking positions that were divisive at least and out of touch in their perspectives.
Biden did not have to give the conservatives all of this ammunition. But he did. Presumably his speech writers decided to focus on gaining points with his base instead of avoiding those divisive topics. Given his poll numbers lately that's not surprising.
But yeah, this was not a unifying speech. It could have been, but he chose to go into topics that were going to split voters along party lines.