This is the process I use to populate a display board with 5 mm LEDs.

Firstly, I recommend that you buy the LEDs in bulk in factory bags, usually 500 or 1000 units, so that you get consistency in the batch. Distrust those deals where they offer a bag of 100 LEDs because they may be seconds. Mind you they will be perfectly fine as single indicator LEDs where you will not notice variations in the brightness.

By convention KiCad assigns pin 1 of the symbol to the cathode, and pad 1 on footprints is usually square. The shorter leg of LEDs is these days the cathode (but ancient LEDs varied) so my mantra when inserting the LED is: short leg square pad. There is also a flat on the circular base near the cathode pin, but I find this difficult to see. Make sure you get this right, otherwise the segment will not work and you're in for painful desoldering. And double check all these steps, trust nobody.

After inserting the LEDs and tamping them down, I place an unpopulated PCB on top and bind the "sandwich" with rubber bands as depicted in the photo. I usually use 4 rubber bands.

Then I pull the legs as straight as I can and then solder all the anode pins (or all the cathode pins, your choice). Be careful not to melt the rubber band with the soldering iron.

I then disassemble the "sandwich". Hopefully you have not missed any pads or the LED will fall out.

Due to variance in pin diameter and the LEDs not being perfectly straight in the "sandwich" some LEDs will not be flush with the PCB. Now is your chance to make them all upright by applying the soldering iron to the pad for each LED that needs straightening. When all the LEDs are flush with the board and orthogonal to your satisfaction, you can solder the other set of pads.

If you are wondering about the THT components in the photo when this design is SMT, this was the previous prototype board. The other THT components were soldered in first, but for the SMT design, there is nothing to solder first.

The display boards are designed to daisy-chain, the top of one board connecting to the bottom of the next board by wire links. There are 8 lines in total. Alternating between front and back links they also provide resistance to flexing.

First clamp the two boards to each other with bulldog clips, or these paper clips.

Take component leads that you clipped off from the LEDs and form them into a rectangular U shape using an axial resistor former like this:

You can also buy such formers from AliExpress, in a different, and IMO more logical, shape. If the gap is short, pliers can be used to form the wire.

Insert half of the links (4) from the front, secure them to the boards with sticky tape, and solder on the back side. Then insert the other half from the back and solder on the front side. It's important that the links are flat and straight so that they resist stretching, and thus flexing of the boards.

Now you can remove the sticky tape, bulldog clips or paper clips.