Rewiring outlet from half-hot to full-hot
The short wires are called &quot;pigtails&quot;.
As you gathered, it was important that the two black non-pigtails remain connected to each other irrespective of the receptacle.  You can use the 2 screws on the receptacle as an ad-hoc splice block... but that doesn't work if the tab is broken. The purpose of the tab is allow you to use it as a splice block instead of for separate control.
You can't buy a new tab.  You have 2 ways you can fix this.
1. pigtail each socket separately.
Leave the two pigtails on it, but simply move the red pigtail from the &quot;red wires&quot; wire-nut, to the &quot;black wires&quot; wire-nut.   You can still do that if you saved the bits.  Or you can buy some loose wire and cut some bits if you have a stripper.
Putting 4 wires on a wirenut is a little challenging, and you'll need red or tan wire-nuts, not yellow ones.
There's no way to replace just the tab.  To salvage a tab-broken recep, just pigtail both sides as above.
2. Replace the outlet
You can &quot;get the tab back&quot; by replacing the whole receptacle - but given the DIY repair boom, prices are going crazy, and you could pay as much as $1.25 for builder (cheap) grade... or $4 for the vastly better quality spec-grade ones.
In fact the spec-grade ones allow 2 wires under each screw, so the builder wouldn't have needed to pigtail.