I don't see many arguments as opposed to preferences or personal statements so far,
but a big one is that local businesses tend to be more accountable to their local communities.
This may mean that they aren't going to necessarily shit where they eat, maybe they will source more things locally and not need cross country supply chains, and hopefully they will hire locally as well. When I think of mom and pop shops that have been in a place for decades, this is what I think of. If I have a complaint, the manager may be the owner, or the owner may be in regularly and I can have a talk with them. If there is something really important to the community that they live in, hopefully theyll have the same interest as me in preserving it. Because their HQ is not located thousands of miles away, they are typically more integrated into the community, and you can put a face to the person who runs the business themselves, instead of just a vague HR hierarchy. They can also respond to changing business needs in a more direct way, aren't necessarily held up by red tape in decision making, etc
Should you always help local businesses?
No. For example, in the city I spent the majority of my life in, before the pandemic we had a spade of gentrification where tons of hipster restaurants, coffee pubs, whiskey pubs, and other gastros appeared in a short period of a few years. I never liked these places, always found the things there to be over priced, and they basically did nothing to really give back to the community other than stereotypical liberal poetry reading nights (whereas several other older businesses I knew of helped with football leagues, hockey tournaments, town halls for schooling events, etc).
When covid came around, some of the hipster businesses, which were local businesses, closed down. When BLM came around, they championed them, even while they were tearing apart the other, less affluent, side of town.
Do I give a shit about these local hipster businesses? No. Do businesses have to give back to the community in a charitable way to be valuable? No. The important thing is really that they are run by nice enough people, offer a decent service, and them being there makes the community a better place because of said services they provide (and no, more avocado toast is not a service that makes your community that much better).
There's definite plus sides to helping local businesses, but in a nutshell, rub the hand that rubs your own back, and if you have good local businesses, definitely do not let them get eaten up by larger chains, or yuppie cafes, if times ever do get hard. You will regret it later, if you do so.