Schwab International is the way to go - it's the only thing they'll let you open if you tell them you live in Germany. That's how I got started with Schwab when I came to Germany 7 years ago. There are several important things you need to know:

The tax treaty will govern whether any dividends you earn in a 401k or IRA are taxable in Germany. Not all of the tax-advantaged accounts in the US are given the same treatment. I'm not an expert, but I think Roths would not be seen as tax free.

In general, using your US address doesn't matter to the Finanzamt. You are taxed on worldwide income. But normally your broker will not like it if they find out you are living at a different address. This is about their sensitivity to compliance more than anything else.

As an EU resident, you are heavily restricted in the kinds of new investments you can make. Let's assume you have those invested in ETFs. Schwab should gladly take a roll-over or transfer of any existing accounts, and they should let you hold existing shares. But you will normally NOT be allowed to buy any new shares. It also means that your dividends will be credited to cash, you cannot have them automatically reinvested.

There is an EU regulation that demands specific consumer disclosures for funds like ETFs - and the shares you hold in your US IRA/401k are not compliant. That means they are not for sale to an EU resident. Instead, brokerages offer what's called UCITS compliant funds that can track the same index, but are listed on EU exchanges. Perfect - unless you are an American citizen. The IRS considers these funds foreign (even if it's an S&P fund) and the tax complexities are extreme. You can google PFIC to get an idea of this. Nearly everyone wants to stay far away.

There are some workarounds:

1. You can freely purchase single stock shares. I'm not really a stock picker, but I spent a lot of time in the past years buying B class shares of Berkshire Hathaway - which is about as diversified and reliable as any single company could be.
2. Don't transfer an continue using your US address. This might work, until it doesn't. As mentioned before, to pull this off, you have to hide the truth from your broker. Maybe not illegal, but still risky. They could freeze your account and force a sale/transfer anyway.
3. Trade OPTIONS on the underlying ETF you want to own. Done correctly, you will pay for, and get shares assigned to you, but you won't have bought them directly. This is a really viable path. Now, options trading is super complicated, and if you're a new investor, you need to really spend some time to research it. I wish I had figured this out 7 years ago, I would have been buying lots more shares this way. Spend some time to research "cash secured puts" if you want to explore. The main limitation is that you have to be ready to pay for 100 shares of whatever fund you want. That normally means regular monthly investing is off the table, you'll have to build up enough cash and buy in larger tranches. Consider Schwab funds like SCHB, since they tend to be much cheaper than the other popular funds from Vanguard or iShares.