I'm actually just starting a business using Stripe for handling payments, and this makes me very concerned.
Is there an alternative service one could use? Is there a way not to give these shady companies power to steal all my money?
For local businesses, perhaps your local bank has a better solution? They will be cheaper (if they use networks other than Visa/Master), and often has zero fees when you withdraw money to a bank account.
Here in my country, for a business that only caters to local customers, I pay about 2% for payments with no fixed fee and no fee for withdrawals, which happens daily. I can directly call the person who handles my queries, and charge backs have never been anything but a few quick clicks with no mind games. All I do is redirect the user to the payment page, and validate the payment upon arrival. The UX isn't as good as Stripe of course, but many of the locals are quite used to that UI anyway.
We see this in many circumstances: centralization and scale are great for efficiency, but bad for resilience.
There are many alternative services listed in other comments, but to be clear, every single one of them has the power to do exactly what Stripe did.
There is no way to accept credit card payments without accepting that power to freeze suspicious incoming payments, simply because fundamentally credit card payments are not final and can (and often will) be reversed afterward, and if someone promised to never freeze merchants' funds, every fraudster in the world would come to try out their services, bankrupting them in the process.
But; I have run 4 different businesses on stripe over about 6 years and have never had a serious issue. Yes there have been ups and downs but even with no phone number I would have to rate their support head and shoulders above most. Maybe 24-36 hours for a response, but we always got one, and in most cases it solved the issue.
We are still with them.
Mollie for one is complete garbage.
It comes in really, really handy when your tiny 1-person company outlives your giant multinational payment provider. Happened to me a few times now, I'm on to my 3rd (4th?) provider in 20 years. There's a tendency for Borg-like consolidation in the payments industry (eg Avangate acquired 2Checkout, then were bought by Verifone). Anyone who lived through the Digital River era will know what I mean.
If you haven't done that code abstraction, imagine that you're suddenly unable to take any payments or make any sales and you are unable to do so until you rewrite all your payment processing code. That can be a long outage.
Having a backup is also useful when a customer's card gets declined by an overly zealous fraud screen at one provider. You can give that customer a "promocode" that changes the logic of your checkout flow and redirects processing to a different payment processor.
And having different processors active also lets you add different payment options (eg maybe your primary processor doesn't support Alipay or Pay With Amazon). But in practice that turned out not to be a good reason for me, Paypal/Visa/MC/Amex/Optima covered almost 100% of customers anyway. For the tiny handful that use checks I can handle that manually.
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