I have a one of the classic Ti cones, and I really like to take advantage of the "tri" fuel nature of the cone.
Though, I don't usually use wood for my goto fuel, but rather, use wood to create "surplus", so to speak, with the fuel I've packed. What I will do is bring *exactly* the right amount (or even slightly less) esbit or alcohol for the number of hot meals I've planned. Then I will try to use wood to boil a meal or two right from day one when the weather and trail conditions allow. These one or two boils will create a surplus in the fuel I have packed, creating some buffer for emergencies, "emergencies" usually being the desire for an extra hot chocolate or peppermint tea at night.
The only downside is that my fuel weight doesn't decrease right away, but really, at about 0.5-0.7 ounces of fuel for a meal and hot drinks, it's not a significant amount of weight. The ultralight ethos of returning home from a trip with absolutely no food or fuel remaining has never appealed to me, mostly because my trips tend to have some degree of off-trail unpredictability, and using wood allows for me to create that buffer at the start of the trip, without packing extra fuel, rather than having to rely on wood at the end, when weather or trail conditions might not be as conducive for wood burning.
I also like to use wood for simmer or cooking foods that take longer than what is easy with alcohol or esbit. In the picture below we are frying vegetables in butter. The vegetables and aluminium foil were packed ahead of time at home, and were cooked on the first night on the trail. I've also used a frying pan placed on top of the cone a few times while burning wood, and it works alright, but you just have to be careful not to bend the lip of the Ti foil when you move or seat the pan.
I also use esbit a fair bit. Although I find that both esbit and wood leave residue, I've found that wiping the pot and cone down quickly while the residue is still warm gets rid of most of it it right away. I try to rub them in the dirt/moss/duff first, and then follow up with a 3 x 3 inch piece of a sham-wow towel.
I've used my Ti Goat 900ml pot many times directly over green pine branches that give off a huge amount of sticky creosote and sap. I have a MYOG reflectix cozy that I use to cover the pot that keeps the sticky mess from getting everywhere. Usually the pot goes right in the cozy from the stove, so the cozy gets pretty sticky on the inside.
I give the pot and cone as much of a clean as I can in the field, and a very vigorous scrub with a steel pot scrubber, dish-soap, and sometimes vinegar when I get home (I'm not actually sure if the vinegar helps, but it makes me feel better). After quite a few years, the cone and pot are discoloured a bit, but not at all sticky.
Burning wood in the cone seems to require a fraction of the fuel than using a normal fire with a ring of rocks, and I find it much quicker, but I don't use wood as my primary fuel simply because I don't feel I have to. Esbit and alcohol are so (relatively) inexpensive, and light that I really don't see a reason not to bring them.
Here is the picture of roasting veggies on the cone: