Back in 2018, it was usually only premium models that had BT. They all have it now. You can buy little BT audio repeater units on Amazon for a few euros. I used one with my previous Sony TV which didn’t have BT. You can usually connect to it via a headphone socket or RCA sockets. Some support digital optical (toslink) connection as well.
As I said , I have tried plug in connectors. I have to use optical output in order to keep sound on tv speakers. I have tried 3 with no success. One didn’t work, one cut off immediately after connecting and one had volume so low I could still hear the tv speaker. I just use subtitles all the time now.
Yes, some TVs turn off speakers when you use the headphone socket. Some also turn off the optical output which is a pain. I’m surprised you couldn’t get one of those cheap BT audio transmitters to work. Some don’t work with speakers/headphones that require a pin, but most don’t. My Sony BT headphones don’t.
The problem with BT is that it can introduce significant latency (> 100ms for the older protocols) which is quite noticeable.
Sadly BT is the only way to get sound from my TV to the amp & speakers I wish to use with it if I wish to be able to use the TV’s volume control.
Again, some TV’s have serviceable speakers such as the 2015 Samsung that is installed chez nous en France - some, like the 2021 Samsung UE55AU9000 I have installed chez nous en Angleterre have speakers which make the one in this
sound good.
Anyway, I digress - I have a choice of visible loss of A-V sync or having to find a DAC with remote control volume. For now the loss of sync is just about the edge of what I can percieve for voice so it’s not too annoying.
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So, no TOSLINK or RCA audio inputs on your amp ?
Edit: I found out pretty much by accident that my TV volume control controls my amp connected by TOSLINK. Didn’t know that facility was part of TOSLINK.
The amp is an Audiolab 8000A - it has RCA inputs galore but I’m not certain TOSLINK had been invented when the amp was designed.
The problem there is that the TV does not have RCA out. Modern sets don’t by and large.
I used to get around this by using the headphone output but this TV doesn’t have one of those either.
The DAC, has USB, BT and TOSLINK inputs (it’s an FX-Audio DAC-6X MkII). Doesn’t have remote control for the volume, however.
The TV can connect to Bluetooth, optical (TOSLINK), ARC/eARC and I think it can send audio using a proprietary Samsung protocol over t’ Wi-Fi.
The catch with optical is that the TV remote does not control the output level. GoK why given that it will do so for BT connected devices. Not sure about ARC, haven’t explored that yet, though I believe the idea is that it can, at least when connected to a sound bar.
Complicated 
It’s weird that my new LG TVs handset volume buttons control the volume of my TOSLINK connected amp, just like they control the volume of my BT connected headphones. Not sure exactly how it does that.
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BT has AVRCP, HFP, A2DP, HFP and VCP (thanks Google AI giving what appears to be a sensible answer for once).
HDMI has CEC
TOSLINK does not, as far as I know send a “protocol”, just raw audio as S/PDIF. “Volume control” can, of course, be achieved by altering the sample levels.
Just looked it up out of curiosity and LG TVs can change volume on TOSLINK connected amps using something called ‘LG TV Sound Sync’ which must be some sort of OOB data channel. I think the amp must of course also support this and so my Cambridge Audio amp must as it definitely does change the audio level on my amp.
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You can have independent volume control on TV, soundbar and BT headphones, mais c’est compliqué.
Feed the soundbar via HDMI and feed a separate BT transmitter with the optical digital audio output. There is a risk of conversion delay putting the video and BT audio slightly out of sink this way though.
Thanks. As I said, tried, failed. Only so many times I can be bothered ordering and returning items to Amazon. 
I don’t need a soundbar, my wife can hear the tv ok.
As it happens, my hearing seems to have improved over recent months , surprisingly.
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It’s probably just a fancy name for the CEC standard mentioned above. 
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CEC is for HDMI not optical. This seems to be an LG proprietary thing that LG optically connected soundbars/speakers and some third party amps support. I just happen to have an amp that supports it.
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Ah gotcha - I assumed it was HDMI… my mistake!!
This 
FWIW I find LG’s WebOS better than Samsung’s Tizen, which was very slow.
But then I hardly use any features of it - I switch the TV on, watch YouTube or iPlayer, and that’s pretty much it.
It talks to my Sonos soundbar automagically via E-ARC, so I am happy.