I don't think you can use the word "audiophile" and budget together.
Since you have the base system, I would probably invest in a better, cleaner amp. I would say the shaker amp is boarderline, so I would think the base amp sucks even more.
Or audiophile and Motörhead
just kidding to each his own.
But yes I agree, the speakers are the weak link and easiest / cheapest upgrade to most factory systems, followed by the amp. I did all 4 of my speakers (also shaker 500) and got great results, especially with the fronts. I am still tossing around the idea of a kenwood kac8405 amp, or similar. I'm not looking to shake windows or break the bank, just clean it up at higher volume.
Speaking of which, if you add enough power you will start vibrating door panels, etc which after spending hundreds or in excess of a thousand can be damn frustrating. Then you'll need sound dampening materials, which are not cheap.
My advice speakers first optionally with a new amp at the same time or phase it in. If youre not looking to wake the neighbors, Get high efficiency speakers (~>90db rating) so that a lower powered amp can push them into their optimal operating range