but I can understand your dilemma.
Started weaning a couple of weeks ago (so not exactly an old pro), so for what it's worth from my limited experience
BLW - pros: great for them to take a healthy interest in food; limited prep if they're eating with you; possibly less face caked in mush, independence, no equipment/jars etc
cons: at first they drop everything on themselves and the floor; they don't eat as much initially; they can be a bit slow, not great if you're in a hurry for something e.g. trying to get out in the morning
Puree - cons: spoon-feeding, preparation/thinking ahead, requires some equipment, messy messy stained clothes and faces, they'll grab the spoon and flick globs of puree, slightly more unpleasant clear-up
pros: portion/intake control of some kind, cute little ice cube portions, some of it's easy (e.g. mashed banana), easier to swallow/digest for weaner-starters, lots of combinations - scope for odd variety - courgette and blueberry if you will.
Whatever you do, they'll gag and choke at some point. I'm introducing stuff as puree then trying same flavours as finger foods for a bit of novelty.
So, on balance, I'd aim for a mix depending on the environment.