Yes, your server as described has to be PCI compliant. However, most processors support modes of operation, different than you describe, which will limit your scope*.
- My client sends the credit card information to my server
- My server forwards this to braintree (or paypal) and I get back a credit card ID token
If the credit card data is transmitted, stored, or processed by your server, then you server is in scope, and you are subject to a reasonably large set of PCI compliance such as SAQ C or SAQ D.
When such a technology is in use, and insulating your systems from any direct involvement with card data, then your compliance requirements drop down to SAQ A or SAQ A-EP.
You should ask your processor "What offerings do you have for tokenization that will allow me to keep my servers out of scope and reduce my PCI compliance requirements?" (If they say "huh?", find a new processor.)
[if so, is it] actually possible to have a payment system that does not
require a user account at the custodian.
No. If you are allowing a processor to handle cards for you, then you must give them a bank account for use in transferring funds to (e.g., purchases) and from (e.g., chargebacks) you. And when they want the bank account, they're going to want all sorts of tiresome information like who you are. The processor is, in fact, sponsoring you into the card network. They are liable for your debts if you behave poorly.
(And that's true whether they're managing the actual card data for you (tokenizing) or just processing transactions... Credit card numbers are not useful without processing by the card network; you can't charge them without a connection to the system, and the system isn't designed for anonymous members. If you want anonymity, charge in Bitcoins or Simoleons instead of credit cards).
*If your business is accepting credit card payments, you're bound by PCI, even if you don't touch the actual credit card data yourself. The best you can do is reduce your scope - if you are subject to SAQ A, for example, you'll have ~15 requirements instead of ~300 for SAQ D.