If I had an out of warranty Honda, I would probably stick to an independent mechanic. I have an in-warranty Honda though. My dealer is so-so. By which I mean, they have fixed the (significant) problems that came up, and not destroyed the engine/car in the process, although there were some glitches. They don't overcharge for oil changes, nor did they weasel out of the warranty when I needed it.
On the other hand, I also have a 30 year old German sports car and limited ability to work on it myself (not having a house/garage) and I felt an independent European car specialist was taking advantage of me, so I tried a dealer of the marque. Keeping in mind they only want to replace things and charge incredible rates, it still seems like the best option for some repairs, to get it done right. When I had a fuel leak, I was happy to have the dealer replace the fuel pump and filter. Maybe somebody intrepid who charges half as much per hour could have just disassembled it and changed some seals, but how much labor is that and do I trust it not to result in setting my car on fire? I certainly wouldn't trust myself to do it.
Sometimes you want OEM parts and sometimes it doesn't matter so much. I'm going to continue to choose what to do on a case-by-case basis.
If you have an unusual old car, a lot of independent mechanics may be horrified at the prospect of working on it, plus the factory manual is out of print plus there's a very limited number of specialist mechanics. So, I'm not saying the cliched advice about avoiding the dealer and going to a good mechanic is completely unfounded, but it's of limited use if you haven't found a good independent. The good mechanics only survive because they're cheap and attract cheap customers, and don't specialize too much, so I say to myself - how can I expect them to be better than the dealer? If I was an employee and I was any good, why wouldn't I work at the dealer? Lots of dealers are terrible, but I infer that independents operate under really harsh constraints. The fact that their labor rates are so much less makes me paranoid both they are padding their hours and that they can't retain good mechanics.
As far as a battery clamp, I can order anything I need mail order, and I can compare mail order prices to the dealership and see if the markup is less than shipping costs. I can choose an OEM part or not. This really is just a separate choice, not a factor in deciding to go to the dealer. My Honda dealer will charge you a ridiculous price to put in a new cabin air filter, which costs almost nothing and is trivial to replace. But that doesn't mean I don't trust them to work on my car at all. Just that I'm careful to double check whether any given thing is reasonable.
But also, a major metro area may be different. I live in a three-city area that has about a million people combined, so it isn't in the middle of nowhere, but it's not like Seattle.