Do you need to be listening while recording? There are numerous options, including for the guy who passed on the BHK, due to no tape outs, unless you want both inside and outside playing at same time.
Setup one input for HT pass through, or do what I do. I have a separate pair of cables going to my ADC. When I want to digitize a LP, I switch cables at the phono preamp. Depending on your phono preamp, this could be easy, or a hassle. My phono preamp has a separate battery power supply and the phono preamp is about 5" x 7". It sits on the top shelf, below my TT, which sits on a Target wall mounted shelf. So very easy access. Or if you are using the balanced outputs to to your amps, just run a pair of single ended cables to the ADC, just don’t change levels during recordings. I’m pretty sure you can use both outputs at the same time. That way you would not have to switch around cables.
Not knowing your ADC, or phono preamp, there may be other options. I’m assuming if you are using fixed outs, that your ADC has a level control. Mine doesn’t, so I actually run the .5m from phono preamp to a pot, then to the ADC. I use Vinyl Studio, with Q-Up on my TT. Except for editing, when I put the track breaks in, and remove the lead in/outs, which I do on the PCs speakers, or my thirty year old Grado Sr60 headphones. I usually do it while watching TV, with the sound going through the main system. I also tend to do it in batches, record 2 - 7 LPs at a time, then edit them after, I am done recording. I can monitor with the PC or the headphones, but seldom do. Once I have the levels set for the LP, I just keep an eye on the meters, to make sure I am not clipping. It has been years since I used a tape monitor output when recording cassettes, but I do remember having to set levels for each LP.
If you post your other gear, ADC, phono preamp, and what you are recording to, including software, maybe we can come up with some other solutions.