Update: Learned from Koen DC in POST 4 below that there is indeed a LHW agent program.

Unless I have missed something I do not think LHW has an agent program such as Prive, STARTS, etc. If you join the LHW Leaders Club, you will earn points and receive "a warmer welcome no matter the destination. Enjoy Leaders Club member benefits, including upgrade priority upon arrival, daily continental breakfast for two, early check-in and late check-out considerations, and complimentary in-room WiFi with your eligible stays". To use this you can just book directly through LHW, no agent program needed.

Book LHW direct for Leaders Club member benefits or use Virtuoso?

I only have limited experience, but it really depends. If using Virtuoso (not all hotels participate) you usually get the "rate of the day", but then you also get the Virtuoso benefits (and you can still earn LHW points I believe). If you book through LHW you usually, but not always have a wider array of rates to choose from and some of them may be less expensive. Plus, you will still get free breakfast and a potential upgrade as a member of the Leaders Club. I have only booked a LHW about 10 times, so I do not have vast experience, but so far, I have always just booked through LWH as some rate there has proven to be lower cost than using Virtuoso. I personally like the freedom of not having to go through the hassle of using a Virtuoso agent (even if they are usually helpful) as I like the freedom to quickly rebook if a cheaper rate comes along... or just change my mind without having to deal with the extra Virtuoso layer.

I do use Virtuoso when it is profitable for me to do so such as my stay at the Langham in Boston recently. I imagine there are some situations when booking a LHW through Virtuoso would also prove profitable (just none for me so far).