Do US professors/group leaders only get a salary, but no group budget?
Certainly the US and German systems are very different. For starters, there is no US national system for higher education (or for lower education, either). The individual states here all have individual systems of higher education, but funding can be problematic there and can vary with the political climate. Also, many of the top institutions, including MIT, are private institutions, without regular government funding. It is too broad to say that no US professor gets a budget, but generally speaking, in most fields, it would be rare to get a budget beyond salary that comes from general university funds. 

Instead, those fields that require such things as laboratories rely heavily on grant funding from NSF and other, similar, agencies as well as private funding. Grant writing becomes a treadmill. In some places your tenure decision may be heavily influenced by your success in getting grant funds for yourself, your lab, and for student support. 

It is true, however, that some departments in scientific disciplines do maintain general laboratories. Chemistry and physics labs, for example. I know of one cyclotron run by a physics department, but I assume they require lots of grant funding to keep the particles going round and round. 

Many, probably most, graduate students are funded as teaching assistants, paid, modestly, by the university for the services they provide. Research assistants might also be paid that way, but, more likely, are paid by grants of the PI of a lab. 

There is very little secretarial and staff support in most fields in the US unless it is grant funded. The once exception, I think, is that universities have offices that manage grants awarded to faculty as assurance to the funders that funds are properly spent. This office may even be responsible for managing private grants, as has been my experience. It is also likely to offer help in obtaining grants and for seeking out grant opportunities. 

Postdocs and other limited-term positions might be paid out of grant funds or out of general departmental funds for faculty. This would depend on the field. But, I would guess that it leaves an individual some room for negotiation, being less regulated. 

I'll make a guess that Germany, as a society, puts a higher value on education generally than is true in the US. Certainly relatively few tax dollars go to education at any level. This wasn't always the case, but seems to be a continuing and increasing trend.