Both of those GPUs only support hardware decode for H.264 so you could always try the H.264ify browser plugin to stop Youtube from sending video in VP9 or AV1 if possible. Those are currently being decoded in software by your poor CPU which is nowhere fast enough.
Those GPUs have fixed-function decode hardware and I don't know if modern browsers can actually use them for D3D Compositing to accelerate H.264. In their day cards mostly relied on Adobe Flash to accelerate streaming H.264, and HD3650 actually doesn't even support that--HD4000 series or HD3000 IGP (in AMD 700 series chipsets) was the minimum. Not that it matters anyway since Flash is dead.
In any case if you downloaded the H.264 video and played it using VLC with hardware acceleration enabled, Unified Video Decoder 1.0 has a severe bandwidth limit of 40Mbit/s for H.264 which is only enough to cover FHD 1080p30 (30 frames per second). 9600M GT can decode video faster but only in particular framesize resolutions (a problem fixed in VP4), however 1920 horizontal resolution of FHD is one of the resolutions it can hardware accelerate. So yes, 9600M GT may play video faster, but only if it's H.264 and one of the supported resolutions.
If it's a H.265 or other video then it will revert to software decode so you'd have the same problem as now if the CPUs are comparably slow.