I rarely see pigtails to join the outlet line/load connections in any of the renovation work I've done. I think it's just easier for the electrician to wire both parts of the outlet, rather than getting a small piece of wire, twist, affix nut, and attach to the outlet. It also takes up less space in the j-box.
Of course the line/load are pigtailed in other j-boxes (e.g. lighting fixtures and switches) since they don't have two screw terminals. And you would pigtail the lines before a GFCI if the next device shouldn't be protected by that GFCI outlet, but otherwise you would always use the line/load terminals in a GFCI.
All you need is a box with enough space. Using a box that you actually need anyway and sizing it to have adequate room for the wires and devices is both simpler and less expensive, generally.
I also don't see a point to removing the existing GFCI - just run everything new off its load terminals, you'll not regret having an outlet there at some point in the future. Put a box extension ring on it to bring it out for ease of conduit connection, if you like. There's no need to add any more GFCIs to the circuit - if you have another circuit, sure, but if a GFCI is the first thing on a circuit, and everything else is on its load terminals, everything is GFCI protected, at no extra expense.
The three gang has (countable) 3 hots, 3 neutrals, 1 ground (or 0 ground, depending on your conduit), 3 yokes (that count for 6 wires) If you are actually using conduit, yay, no clamps to count.
12 or 13 wires to count (wires that don't leave the box are free - not counted) presuming 20A circuit and 12 Ga wire, 2.25 cubic inches per "count", minimum volume 27 or 29.25 cubic inches depending if you have ground wires (all of which count as one, none of which are needed with metallic conduit such as EMT.) You can always go bigger if you want more space to work in, but it costs more, usually.
Scanning a few 3-gang boxes, even 2-1/2 inch depth ones have more than plenty of space for this use. 3-1/2 inch depth ones are even more generous.
Best Answer
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 & 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.