I'm not a client of a private bank, but I have friends in the industry so i have a little bit of familiarity.
For <1,000,000, you would not be eligible for what is generally called "private banking". Usually, that's for 8 figures in liquid assets or more, although exceptions are made for people likely to be there in the next few years (e.g., non-liquid startup founder of a unicorn).
With a "private bank", you'll have access to financial experts well-versed in the tax consequences of your investments. They'll be familiar with trusts and other things like that. They might give you access to exclusive deals. And you'll have a phone number that you can call any time of day and speak to someone intelligent (at least compared to the person answering the phone at your local Chase branch). Most importantly, you'll have people who can make you customized deals. For example, when Jeff Bezos wants $500m to spend on a yacht but doesn't want to sell his Amazon shares, his bank will happily give him a loan on good terms for that. If you own a Picasso and want to get a cash loan with that as collateral, your banker will at least consider that request. For most people, the only asset they can borrow against is their home. But at the private banking level, they are more likely to consider other assets. They will at least humor the request to look into it.
What you are talking about is more of a "financial advisor". 1% is typical. If you are simply interested in investing in index funds and have the discipline to not touch your funds when the market has ups and downs, then you may not need/want one. But I would pose the question to a financial advisor and see what services they offer that might make the fee worthwhile for you. Typically, things like tax planning start to get more important when you cross the $10m mark and inheritance tax becomes a thing.
With regard to investing, an "all-world passive index tracker" may not be what you want. I'm not sure where you are located, but assuming it's the US, you might want to allocate a good amount of your money to US indices. The problem with all-world is that you are most likely not located "all over the world". You are most likely tied to the US and so you probably want assets closely tied to the US economy, which is where you will probably be spending most of your money.