In the past I've always pulled the main root and leave the little ones in the ground.
A lot of people seem to dispose their plants off site which is a shame. I couldn't bring myself to do that but I did make a separate compost pile for tomato plants, and a few times I have hauled them off into the woods to rot in their own spot. They produce a ton of biomass for composting, I mean tomato plants are huge producers of vine no matter how well your fruit do.
In fact I haven't seen any detrimental effect of growing tomatoes in the soil where their own roots decomposed. And I have let volunteers to grow in a compost pile that contained some tomato waste, and they weren't bothered by that either. So I'm not really convinced about the mythology of tomato plant disposal. The scorched earth approach doesn't seem justified.
In large commercial operations the recommended practice is to till the plant residues into the ground and then rotate to a different crop for one year. Basically just about all tomato diseases are destroyed when the plant material is completely decomposed.
Another safer way is also easy to do - make more than one compost pile. Have a pile where you incorporate the tomato vines and a separate one for garlic/onions waste. Then you can feed them to each other and not worry about any carry over of diseases etc.
Composting may be the best way of adding carbon to the soil instead of releasing to the atmosphere, according to one 19 year long study. There is a big footprint for sending vines to the dump or burning them, which could be sequestered instead. Even if you keep them out of the veggie garden and just feed them to your shrubs and trees, we could do a good thing by composting our dead plants instead.
