two pairs of bookshelf speakers with at least 4" drivers (ideally 5" or more) at $150 each
You can find decent speakers for less. With some patience, half that much.
And I'm not convinced PC speaker are that great anyway, any reviews comparing them to standalone speakers?
I suspect even normally-priced $80 speakers could be more than competitive.
As I mentioned, nearly all audiophile speakers are beyond the point of dimininishing returns. If you double your budget you definitely do not get double the sound quality.
I'm not in the US so you'll have to excuse my approximations on budget - but popular "budget" bookshelf units over here start at about £50 ($60) for small, 3" drivers using the cheapest acceptable materials (silk-dome tweeters, plastic-coated paper cones in the main drivers, and enough MDF and glue to avoid resonating cabinets) That's not really big enough to deliver full range audio, and there's absolutely no point in buying Hi-Fi separates if you are going to cripple the system with small, cheap drivers that don't sound any better than affordable computer speakers.
At 4" drivers, some of the better (so more expensive) kevlar drivers can put up a decent performance, but the "best buy" entry-level and budget award winners tend to be the cheaper 5" drivers using aluminium/polymer drivers. You can pick these up as old stock, entry-level models from lesser brands for £100 ($115) or so, but that requires shopping around a fair bit. Most of the award-winning speakers in the entry-level or budget categories seem to have an MSRP of $200 and it doesn't take me much effort to find deals for them at $150.
So yeah, you can definitely build up a 5.1 setup using Hi-Fi separates for less than $650, but given that they're going to be undersized, limited-range, basic speakers using paper cones and cheap crossover circuits, they're unlikely to sound any better than a cheap $200 5.1 system or PC speakers, so just buy those instead.
As for reviews and comparisons, check out noaudiophile.com. That guy is a true nerd and his site is mostly about reviewing speakers with the aim of creating DSP correction profiles to compensate for a set of speakers' flaws. He tests everything from silly-expensive to $50 unbranded junk from China. I enjoy reading it, at least, and it's refreshing to find a knowledgeable audiophile who doesn't have their heads up their own arse
I did grew attached to 5.1 output for gaming. I was wondering if I would be sacrificing that with if were go with a home theater route.
Nah, a proper AV receiver will show up as a 5.1 device in Windows 10 when connected via HDMI. I've tested this on RX 480/Vega, Nvidia 900/1000/2000 series. It works direct to my basic receiver and also via ARC on my TV, and my receiver is good but very much a basic entry-level one: ( Yamaha RX-V383