I am in the second year of running my own wood fired Neapolitan restaurant (well takeaway in these covid times) in the UK. In the UK at least, wood fired is easier and cheaper to install. Gas you need a more specialist flue system. Which needs to have a fan, which costs about £4000 - plus maintenance and electricity. The wood is almost zero maintenance, just a £50 chimney sweep twice a year. The cost of the gas burners also adds a few thousand pounds to the oven cost.
I did really look into gas as you say it is much simpler to use in theory. I think the gas connection cost if you have gas lines in the premises already was about £2000.
I would say gas is safer than wood, it's not going to spit, embers/logs cannot roll out of the oven. At the end of the shift you switch off the gas, the oven is off then and there's no ignition sources left inside/embers. With wood, even an hour or two after using the oven there are live embers, you just seal the oven with the door keeping the embers inside - you could always remove them all each night and safely extinguish them but by keeping them inside you keep the oven hot thus reducing the amount of wood needed to get up to temp the following day. If you ever had to quickly evacuate the building etc. you can easily switch off the gas, if you currently have a roaring inferno in the wood oven you cannot - short of chucking water in which would crack and severely damage the oven.
In the UK we have 'clean air zones' which means you can only use wood ovens from a selective number of producers who have had their appliances certified as exempt for use in smoke control zones, if you don't you can always pay to have the specific appliance tested (for a few thousand pounds) but if it fails the test it would be illegal to use it... so that's a huge risk.
I think a major plus for wood is the consumer appeal, people love the romance and fashionability of wood-fired, it's a major selling point. This is even more so if your oven is on display to customers. The smoke from the wood does also add a small amount of flavour, though this depends on if you dome the pizza much (where the smoke is)/the wood you use/how long you cook for - since Neapolitan pizza cooks in under 2 mins the amount of smoke flavour imparted is very minimal. I would definitely say the appeal to customers of the wood is a far greater factor than the actual impact wood has on the pizza.
Finally, staff training. Wood is much harder to teach and to get to a good level your chef will need months of training. This is the biggest disadvantage. Staff turnover in this industry is really high. On AVERAGE chefs stay about 12 months at any one place. You're therefore going to be spending a lot on training people to use the oven and doing so frequently. Unless of course you already know someone extremely reliable/have a partner who is going to be working on the oven/family/get lucky? It's easy to walk up to the oven, put a pizza in, turn it and cook it nicely. But managing the fire single handed for hours is much more difficult, learning the oven's hot spots takes time. Knowing how to adapt the fire in busy times takes time to learn. Most importantly, you need (and I hate the word for its overuse) but 'passionate' chefs. People who want to learn and do the best.
Not only is using a wood fired oven physically tiring (bending, loading wood, heat, managing peels) but also mentally tiring too. You're always looking at how many pizzas are going in next, thinking is that spot going to be too hot now as although it was perfect 5 minutes ago I have had a big flame so it's now too hot, if you have 5 pizzas in you need to know the ones closest to the fire will cook quickest and not forget about the ones furthest away either - 5 pizzas all cooking in 90 seconds doesn't leave much room for error, crusts can go from pale to black in 15 seconds, if 3 pizzas are all ready at once because the flames are big/temperature high by the time you've removed pizza 1 and 2 from the oven pizza 3 is burned - you therefore need to manage the oven in such a way that staggers the cooking - bearing in mind a marinara will cook much quicker than a meat laden pizza or calzone. If you don't have someone who cares, the results will be extremity inconsistent. Too hot underneath and you have black pizza, too cold and its undercooked. And it's not as simple as just whacking in a big bit of wood if it's cold either. You need to always have a piece of wood in the mouth of the oven so its heating up and therefore catches instantly when put on the fire (instead of suffocating the fire for the first minute if it's cold). Too big of a flame and the pizza might burn on the outside and be undercooked underneath. You need to adjust cooking also based on the dough - less ready and a smaller flame/lower temperature is needed. For this reason I would recommend extremely consistent dough in terms of proofing so that there's one less variable to worry about.
Wood fired ovens are also easier to damage, if you throw the wood in you can cause cracks, so the chef needs to carefully use a peel to add wood. If the oven gets too hot water can CAREFULLY be used to cool it down but only a tiny amount. This should be done with either a slightly damp (in no circumstances dripping wet) towel over a peel which needs to be moved briskly in the oven. Or a very slightly damp mop. Chucking/spraying any quantity of water or even worse ice is very likely to lead to cracks.
All in all, wood has greater appeal and potential pay off. It's slightly cheaper to install but in terms of training in the long term will be more expensive. I would recommend wood if your local laws allow it and you're up to the challenge of a more tricky skill. Of course like most things in life, if it's more difficult and therefore scarce, doing it well provides bigger pay offs.
I've definitely rambled on enough now, if you have any questions please ask I will be happy to advise. Final point: In the UK at least wood and gas is about the same cost per unit of heat currently. Gas however is cheaper in the sense that you don't have to pay staff to accept wood deliveries, move wood and potentially chop up wood if it is too large, nor do you need space to store gas as you do with wood.
