Because I have liked Audio-Technica products quite a bit in the past, after posting the initial review I decided to do some more delving, and contacted the company. Their representative directed me to a video about setting up the device, and so I dove in again. That's because not only does it demonstrate the pairing process much better than is described in the manual, but one of the most valuable things mentioned in the video is that if you have trouble setting up the Bluetooth connection (the way manual describes), you might instead want to try going into pairing mode with the turntable first, rather than the speaker first as the manual appears to suggest. I tried that and had substantively more success. There still were a few hiccups, but overall it went much more smoothly.
As a result, I've rewritten the end of the review. And for thems what are interested, here is now how the last part reads --
The short version of all this is that I largely like the Audio-Technica AT-LP60-BT turntable. It serves a great function. There were difficulties getting it to work at first - a few small technical glitches perhaps but mainly it seems because the manual is skimpy - and unfortunately that helpful video wasn't embedded on the product page, which would have been an awfully good idea. (You can find it on YouTube, by searching for the product name.) So, after all that, I think the AT-LP60-BT is a nice product that has a lot good about it, just one with the aforementioned caveats. It should work well, but I'd be remiss if I also didn't mention an alternative - particularly since it's one that lets you stay in the Audio-Technica family.
The company sells a range of fine turntables, but for our purposes here they have a near-identical low-end model, the AT-LP60. It retails for $99, while the AT-LP60-BT model being reviewed here is $179. You could get this less-expensive, non-Bluetooth unit, along with a well-regarded Bluetooth transmitter (such as one from TaoTronics - which I haven't personally tested) that sell for around $30. Connecting this transmitter dongle would then give you the same wireless Bluetooth capability. That would not only most-likely provide a much better Bluetooth connection, but also together would cost $50 less than the AT-LP60-BT. (As mentioned, I haven't tested this yet, but I should be getting the transmitter, and if so will do an update later on.) There are of course many other turntables on the market - not just low-end, but high if you'd prefer a more substantial phonograph - for which this would work, as well.
If you don't want to go to the trouble of setting up the devices that way, and prefer the simplicity of an all-in-one unit, the AT-LP60-BT turntable being reviewed here is good and offers a lot of low-end benefits for wireless audio. And happily, I did get it to work. And ultimately work reasonably properly.