The HDMI audio extractor is a great way to get 5.1 surround sound into legacy devices that do not have HDMI connectivity for audio. This is a common solution for older AV receivers which were excellent at the time, but lack HDMI. The Yamaha RX-Z1 for example.
I'm not sure if you live in an area which has such a reselling website like Craigslist, or similar, that people can sell their used electronics on. But, I've seen complete HDMI enable 5.1 surround systems sell for under $200 in my area. Not 'great' speakers, but entry level and usable, along with something like a slightly dated Yamaha, Denon, or similar receiver. They don't pop up often, but they do show up and can be in the price range of about $200.
For people buying off the shelf, it's a bit more, but not crazy expensive. $350 in the USA, can get this system which is an upgrade on almost any speaker bar ever made...
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More importantly, the speakers can be upgraded to MUCH better models as time and budget permits.
The problem with something like a speaker bar, or a packaged Logitech system, is that it has zero upgrade path. If something breaks, the entire system becomes garbage. It doesn't offer industry standard connectivity for modern HDMI devices. It is, at best, a mediocre value proposition for the long term.
A receiver + separate speakers is best because each and every piece offers some level of upgradability. If the speakers aren't great, you can upgrade them. You can add a better subwoofer at any point. You can swap the receiver for a better unit, and your existing speakers will still work. Many people buy really nice speakers and use them for decades, while they change out their AV receiver every few years. But, just to get things rolling, an entry level home theater system allows you to place the speakers around the room wherever you want using standard speaker cable.