If memory serves me right, cookies were designed by Netscape in 1994 before JavaScript was even a thing. They were released in an early beta of Netscape (0.9 something), while Javascript was only added in Netscape 2.0. SSL 2.0 was only added in Netscape 1.0. So the HttpOnly/Secure attributes were not relevant to the first cookie design, which wasn't even a standard.
When the first cookie standard (RFC 2109) was released, the Secure attribute was added. You could argue they missed HttpOnly, but JavaScript itself was highly non-standard and underspecified mess during this period (and for a while later too). Almost nobody was thinking about XSS as far as I can tell, and that term was probably only coined at least 2 or 3 years later (by Microsoft researchers[1]). At 1997, the people who even considered XSS, probably only saw this as an HTML injection issue that can be fixed at the injection site and doesn't require any special protections against JavaScript code.
If you really want to point a finger at issues in the early cookie design, then you could talk about domain matches. Not making the port part of the matched domain and not allowing an explicit way to trigger an exact domain match with a specified domain was a mistake. And the confusing leading dot rules (probably for optimizing a substring match without parsing the domain components) was also a mistake.
But RFC 2109 was replaced with RFC 6265 and now RFC 6265bis (which is not released yet, but is mostly implemented by the big browsers). These RFCs fix most of the big issues we had with cookies, and do not shy away from breaking existing behavior: setting SameSite=Lax as the default and restricting SameSite=None to secure contexts broke A LOT of sites for improve security. The changes made in RFC 6265 to forbid multiple cookies in one header also broke many sites.
An equivalent approach in the JWT spec would be a new RFC that does the following:
1. Forbids implementing Unsecured tokens and removes alg="none".
2. Removes RSA and ECDSA (or at least deprecate them) and make Ed25519 the "Recommended+" signature algorithm. If we allow RSA and ECDSA for compatibility, they must be explicitly enabled with feature flags or some other marker that signals their insecurity and security advisories on their potential vulnerabilities must be attached.
3. Removes the entirity of JWE as it is currently implemented (to be replaced with encryption else that is NOT orthogonal to authentication).
3. Requires that HMAC secrets are specified as binary base64url data. They SHOULD be generated by a CSPRNG or a derived using a safe derivation method (such as HKDF-SHA256 with safe key material) and MUST be at least as long as the "security size" for the algorithm (i.e. 32 bytes for HS256, 48 bytes for HS384 and 64 bytes for HS512).
4. Makes the "exp" claim mandatory to set, and mandatory to verify.
5. Add a section with strict implementation guidelines for libraries, e.g. `parse()` functions that skip verifying the token should have a clear name like `inspect_without_verification()` or `dangerously_parse_without_verifying()` and `verify()` function should always receive a key that is strictly typed to a specific algorithm.
6. Remove or restrict the usage of fields that allow the JWT sender to dictate the keys used for verifying it, like "jku" or "x5c". For instance, the standard can mandate that when implementing these fields, the verifier MUST NOT accept any JWKS URLs that do not match one of the explicitly allowed patterns or X.509 certificates that are not signed by an explicitly trusted CA.
[1] https://www.youtube.com/watch?v=mKAWpFdVcPY
