When the excess contribution is the result of the combined contributions of the spouses but neither contribution, without regard to the other spouse's contribution, is in and of itself an excess contribution, the excess contribution is a shared excess contribution. It is not explicitly attributable to one spouse or the other until an action is taken by the spouses to attribute the excess to one spouse, the other spouse, or in some combination between the two spouses. Because only excess contributions are permitted to be distributed from an HSA as a return of excess contribution, obtaining a return of excess contribution from a particular spouse's HSA explicitly attributes that amount of the excess as being in that spouse's HSA.
With regard to the irrelevance of the investment performance in your wife's HSA, I'm referring only to the legally required calculation (https://www.law.cornell.edu/cfr/text/26/1.408-11), substitute "HSA" for "IRA" since the law requires the same calculation for HSAs as is required for IRAs) of the earnings that must be distributed from your HSA to accompany the return of excess contribution from your HSA. Obviously it would have been better to have less earnings required to be distributed by basing the earnings calculation on the investment performance in your wife's HSA, but that would have required that the excess contribution be distributed from your wife's HSA.
Regarding intermediate distributions, no, I am definitely not saying that a nontaxable distribution used to pay medical expenses is considered to be a distribution of earnings. I'm saying that any regular distributions during the calculation period must be added back to the closing account balance in the calculation of attributable earnings. to determine attributable earnings. See the definition of Adjusted Closing Balance in CFR section 1.408-11(b)(2) referenced above.
It appears that it was PayFlex's mistake not to calculate and distribute the earnings as they said they would on the form. By distributing exactly the amount that you indicated was the amount of the excess contribution, PayFlex is effectively saying that there was exactly $0 gain or loss during the calculation period, which you say is not true. You'll need to contact PayFlex to inform them of their error and have them make the correction. Since there were gains, they probably need to treat the first return of excess contribution as return of only roughly 91% of the excess, with about 9% of the excess still remaining. For example, if the excess contribution was $1,668, there was a 10% gain in your HSA and the gross distribution of a return of excess contribution was $1,668, that would mean that $1,516 of excess was returned accompanied by $152 of earnings, with $152 of excess contribution still remaining to be resolved.
