Nope. Safety Skein does not imply the safety of Threefish. Putting it positively, if someone finds weakness in Trifish, this does not mean that Skin is also unsafe.
The question, however, is quite interesting and relates to other hash functions. Skin uses the Davis-Meyer construction with some modification. MD5, SHA1, and many other hash functions also use this Davis-Mayer construct, and therefore are based on a block cipher. It’s just that in the case of MD5 or SHA1, which block the cipher, they don’t have a name, and I don’t know much research on how suitable these constructions are.
The requirements for a good block cipher and a good hash function are different. It is somewhat simplified if E is a block encryption, and it is not possible to find two keys K, K 'and two messages M, M' such that E K (M) xor M = E K ' (M') xor M ', then E is suitable for constructing a hash function using Davis-Mayer. But for security, other properties are needed as a block cipher E. E would have to withstand attacks with the selected ciphertext, attacks with the selected plaintext, etc.
In addition, if E is a good block cipher, then this also does not mean that it gives a good hash function. Microsoft had to learn this hard way with the hash they used in XBOX. This hash was based on the TEA block cipher, which had a weakness that was not significant for block ciphering but proved fatal when used for the hash function.
To be fair, there are some relationships between a good block cipher and a suitable hash function. For example, in both cases, differential attacks should be avoided. Therefore, some constructive methods used to build good block ciphers can be used to build good hash functions.
Let me also add that some suggestions for SHA-3 are based on AES. So far, I have not seen much support for supporting AES hash functions, simply because AES is already a standard. These hash functions are parsed just like any other SHA-3 clause.