Well bloody nora, it turns out that it truly is a GPU bottleneck! I took the exact same Phenom II system and put a measly HD5450 into and and, sure enough, it's hitting 60fps.
What threw me off is that, despite the HD3450 being a discrete graphics card, it actually uses the exact same core configuration as my integrated HD3300 (heck, they're both even using DDR2 with a 64bit bus).
Speaking of which, I can also confirm that it's not a memory bandwidth issue since, even when using a different motherboard that is virtually identical (including the use of the very same HD3300) except for its support of DDR3 paired with 2xDDR3-1600 and the iGPU configured accordingly to take advantage of that faster RAM, I was getting the same ~35fps as my lowly 2xDDR2-667.
So yeah, the advice would be to either get a more modern CPU that additionally includes more modern integrated graphics or get a considerably faster GPU, try pairing it with your current Athlon and then, maybe later, go with a faster CPU regardless of its integrated graphics (all of my previous advice regarding cheap used LGA1150 CPUs and cheap used discrete AMD GPUs still stands).
Do keep in mind however that AMD CPUs stagnated in single-threaded performance (what Dolphin cares about) until Ryzen came out (Zen-based Athlon CPUs will also work too) which is where my mention of cheap used LGA1150-based Intel CPUs comes in.
What threw me off is that, despite the HD3450 being a discrete graphics card, it actually uses the exact same core configuration as my integrated HD3300 (heck, they're both even using DDR2 with a 64bit bus).
Speaking of which, I can also confirm that it's not a memory bandwidth issue since, even when using a different motherboard that is virtually identical (including the use of the very same HD3300) except for its support of DDR3 paired with 2xDDR3-1600 and the iGPU configured accordingly to take advantage of that faster RAM, I was getting the same ~35fps as my lowly 2xDDR2-667.
So yeah, the advice would be to either get a more modern CPU that additionally includes more modern integrated graphics or get a considerably faster GPU, try pairing it with your current Athlon and then, maybe later, go with a faster CPU regardless of its integrated graphics (all of my previous advice regarding cheap used LGA1150 CPUs and cheap used discrete AMD GPUs still stands).
Do keep in mind however that AMD CPUs stagnated in single-threaded performance (what Dolphin cares about) until Ryzen came out (Zen-based Athlon CPUs will also work too) which is where my mention of cheap used LGA1150-based Intel CPUs comes in.
Dolphin 5.0 CPU benchmark
CPU: Xeon E3-1246 v3 (4c/8t Haswell/Intel 4th gen) — core & cache @ 3.9GHz via multicore enhancement
GPU: Intel integrated HD Graphics P4600
RAM: 4x8GB Corsair Vengence @ DDR3-1600
OS: Linux Mint 20.3 Xfce + [VM] Win7 SP1 x64
CPU: Xeon E3-1246 v3 (4c/8t Haswell/Intel 4th gen) — core & cache @ 3.9GHz via multicore enhancement
GPU: Intel integrated HD Graphics P4600
RAM: 4x8GB Corsair Vengence @ DDR3-1600
OS: Linux Mint 20.3 Xfce + [VM] Win7 SP1 x64