I have also read that Sapphire Rapids CPUs can accelerate Stable Diffusion, although I'm still not quite sure how...
I think this is because of native support for AMX instruction set (evolution of AVX-512), unlikely IAA or QAT accelerators are involved (and DSA / DLB accelerators are completely for something else).
...Intel Xeon w3-2423 is available within budget if a similarly-cheap motherboard is acquired (thus, no need for in-demand Gigabyte). Do all LGA 4677 boards support all processors? I didn't think so, but then I thought I saw something somewhere that said they were interchangeable between workstation and server board...
Probably another thing to consider is memory bandwidth - W3 will be significantly constrained.
There are desktop/workstation CPUs and Server CPUs, those are hardly interchangeable and different chipsets (motherboards) are designed for different CPU class ( {a} desktop or workstation - vs {b} Server )
Sapphire Rapids are based on Golden Cove microarchitecture - same as P-cores of 12th Gen Desktop model CPUs, therefore in your comparison to i5-12400T (T - have artificially lower specs) is quite fair.
...then I thought I saw something somewhere that said they were interchangeable between workstation and server board...
This is unlikely to be intentional, I would say say it happens by coincidence (same platform architecture after all), not worth relying on this functionality as it may go away or exhibit some stability issues, I do not expect much of a warranties by suppliers for such support.
...However, it may be a moot point if the w3-2423 isn't even capable of working well with multiple high-spec GPUs at once...
Again - severely constrained memory bandwidth (less channels and reduced frequency), since you've mentioned "Multiple GPUs" then also probably taking into account reduced number of PCIe lines on w3 (so if you have all-slots MB - not all might be operational)
Intel Xeon w3-2423 is available within budget if a similarly-cheap motherboard is acquired (thus, no need for in-demand Gigabyte)
Depends on your semantics for "budget", some full-scale SPR QS with all features are available as low as $1500 and dual-CPU boards are going for around $1000 (plus extra cost for DDR5 RDIMMs), although I haven't seen boards with all PCIe lanes implemented in E-ATX form-factor (to put into regular desktop/workstation case), my dual-cpu SuperMicro X13DEI (with single 8461V CPU) despite fitting the case - has lots of PCIe lanes missing (or alternatively X13DAI if you want couple spare PCIe lanes and integrated SoundCard so you won't have to waste another {even x4 bifurcated via splitter cable/board to mount somewhere in a case} slot for it {unless it's USB})
. Looked at the ASUS W790 Ace & SAGE SE boards they look pretty good, although I need a remote console in case things go weird for some reason so maybe not a good option. 