Regardless of whether tax is proportionate or not, the wealthy will always be paying more because 10% of 1,000,000 is going to be more than 10% of 10,000. I personally believe that it is not the tax rate itself that is the problem, but exploitable loopholes that make it so that anyone does not have to pay the full tax amount they are quoted to do so.
However, at the point at which tax rates functionally become too high for any citizen to manage, that citizen may decide to move and take their business elsewhere in favour of somewhere that is more financially favourable to themselves. Of course, that decision is easier to make for someone with money than someone without, but increasing tax rates/brackets too much could incentivize a flight of important capital. Tax the rich, but don't tax them so much that they'll leave and you'll lose the benefits of their taxes.
There's also the question of what rich really means. Is a rich person someone who has relatively more money than you? Is someone who earns $100,000 a year in a city like San Francisco really rich when you consider the high cost of living associated with living in a city like that? Should some 'rich' people get subsidies or deductions on taxes if they are demonstrably contributing to society in important ways such as employment of a large portion of the population or significant charitable donations?
It's interesting to note that countries/states/cities have a lot to gain from keeping wealthy people as taxpayers and residents of their region. If they're paying more taxes, that's more money that can be used towards spending on public services and keeping the region afloat. If they are at the head of a large company that's headquartered in that particular area, that gives employment (and wealth) to many other people.
Sometimes, there can be severe and unintended consequences when tax rates, or methods of taxation, are increased on the rich. In France for example, the system of wealth tax was changed in 2018 in favour of a tax on real estate. Previously, households with more than €1.3 million in revenue could write off 75% of charitable contributions (up to €50,000) off of their taxes. To simplify the situation, under the new law, only half as many people could do the same write-off, and it led to French charities losing a lot of money. France Générosité, which represents 97 charities, saw a €200 million decrease in charitable income in 2018 alone, and only amongst their represented members, and cites the wealth tax change as one of the main contributors of this decline.
In this case, trying to tax the rich more by changing taxation rules ended up having a severe impact on charities, many of whom had to cut their services in light of this lost income. Everything is so connected in our society that it's easy to say "tax the rich" or "eat the rich" without considering the importance of a rich person in society.