LMSs were generally superceded by server technology, but often used in conjunction with FlexiCarts, a Sony DigiBeta based machine similar to an LMS. It was however a simpler, narrower machine which held far fewer tapes and usually contained one or two transports as opposed to five or more. The robotic arms when up and down a single rack of tapes rather than up/down/left/right across several racks. You can still find Flexicarts in tape ingest areas of playout centres, but AIUI they too are becoming obsolete now.
Yep, Sony stopped making them a few years ago. Automatic tape loaders aren't really required any more in playout areas as stuff is manually cached onto server once and then played multiple times from there. If it drops off the server it can be retrived from a traditional library and re-cached.
The main use of Flexicarts in recent times is for for archiving - a DVCAM one can hold about 140 (3 hour) tapes giving a big, easily accessible archive.
The BBC had one LMS in BBC World transmission which was used until the early 2000s. BBC 1 and 2 used a Panasonic multi-tape/transport system called a MARC (in fact they had two MARCs). ISTR Thames Television had MARCs too, though whereas the BBC's used Panasonic D3 decks, Thames' used the analogue M2 format.
Indeed, and I have had the displeasure of working alongside all 4 of those MARCs! They had to be treated with care, bless em. All 'robotic' devices like that had the tendancy to throw a wobbly at times. I spent many years wrestling with Flexicarts too!