Should I use pigtails for pass-through split outlet connections?
Pigtail , pigtail , pigtail, and hook up your new receptacle. The back stab (the push in connection is the largest failure point of any wiring method I have ever heard of but “safe” because when they fail they are inside a box.  Hook your 3 blacks and a short section of wire together with a wire nut then this shorter wire to the gold screw. Do the same with the white wires and connect the short white to the silver screw and you have eliminated the major possibility of a failure if someone plugs a space heater, or hair dryer with a few other devices going also down stream. Is it legal yes it will pass inspection in many jurisdictions but not all. Is it safe , according to the national electric code, but I have seen smoke damage many times and even plastic box meltdowns but no major fires. But those push in connectors are a major failure point. I like back &amp; side receptacles they can connect up to 4 wires each side but cost a couple of $ more. So pigtail with a wire nuts or get back and side outlets. I’d your boxes are shallow spending a bit more on the receptacle is the only way to go and they last longer as they are a higher grade.