Maybe from a Credit Union or other smaller bank, but not a big-box bank.
why the lack of flexibility?
There is a huge secondary market for mortgages through the MBS (Mortgage-backed Security) market. This market consists of millions of mortgages that have sold by banks to other institutions, that package thousands of them together into "bonds" that are then bought by investors. The characteristics of the bonds inside are summarized by tenor, credit rating, etc. so investors know what they're paying for. Throwing 23-year mortgages into the mix would disrupt the models that investors use, making the bonds less marketable.
Since the overhead for making a "custom" mortgage would be relatively high, they just don't offer it to the public. It's not worth their time. A small local bank or credit union, however, might be more willing to accommodate custom needs (or come up with an alternative). However, I would not expect that you'd pay an "interpolated" rate. You may end up just paying the 30-year rate with a custom payment amount.
Just get a 30-year mortgage and make extra payments to pay it off early. Or get a 15-year loan, pay less interest and pay it off even earlier!