Only Carnac the Magnificent can evaluate a book without reading it. Even though you've never actually accessed it, you conclude "n all probability it's a double count".
At the outset, by the way, and as I indicated, Tenney's conclusion (and he makes clear that it's'an estimate) is that the uncounted Union MIA is approximately 4,000, not "5,000".
Let's start with the fact that Tenney's work regarding the respective strengths of the two armies in his 1992 master's thesis earned him a degree and subsequent citation by the likes of Joseph Harsh. Yet you assume that the same guy who turned out that level of scholarship became too dumb to consider whether he is "double counting". In fact, and not surprisingly, he expressly refers to "double counting" issues in this chapter of his book.
In this chapter of his book, however, his primary focus is on Union records, including the morning reports for units that are in the NARA. What he found directly challenges the accuracy of the "6,041 MIA/Captured" in the OR that you're relying on. His conclusion is that most Union army units were using the numbers for "missing" in their morning reports to mean those missing in battle and not necessarily including stragglers during the continuous retreating who also went missing. He located the morning reports of certain units for June 30 and July 10 that recorded "missing" and made a comparison with what is shown in the official "6,041" total. Those units – four brigades and a few individual regiments – show 484 more men "missing" than what shows in their official "MIA" counts. This is because the official count appears to be based on those missing in battle and not those men who went missing as stragglers during the incessant retreating and were captured. A good example is Barlow's 61st NY. The official count shows 20 MIA rank and file. According to Tenney Barlow's morning reports in the NARA show an additional 55 in his "missing" count beyond the 20. In fact, although Tenney relies on the material in the NARA, Barlow's report in the OR specifically states that there were "20 missing on the battlefield" but that "[o]thers fell out on the various marches to and from action, and many of them are doubtless prisoners". OR Vol XI, Part 2 at 69. Tenney points out that 7 men who were missing in the 61st's June 30 report had returned in the July 10 report, reducing the additional "missing" from 62 to 55.
In addition, Tenney looked at several specific units that show "no loss reported" in the official numbers but found primary sources, including reports and other accounts, showing that these units did in fact have losses during the June 25-July 2 time frame, including "missing".
There are plenty of accounts in the OR that show significant straggling by the Union army during these retreats and the capture of significant numbers of stragglers by Confederate units. We know that in war stragglers from a retreating army frequently end up in the wrong hands, while those from an advancing army frequently end up back with their own side. Tenney cites sources from both sides regarding Union stragglers.
In short, we know that the official "MIA" count is wrong.
Regarding those troops captured at the Savage's Station hospital
First, the reports of two hospital captures by Jackson – Savage's Station and then another, much smaller hospital at White Oak - show that both captures included "sick and wounded". That makes sense because Savage's Station was where those initially in the various field hospitals were sent as the Seven Days began. No doubt a large number captured were wounded and it's reasonable to assume that many of those may have been recorded as WIA but there were likely a substantial number who were "sick". One can find references in reports and accounts that during the Seven Days there were "sick"/"unwell" troops who were in/directed to hospital. Swinburne (who you reference), in the pamphlet containing his report expressly stated that at the Savage's hospital on June 26 – before it began receiving wounded – the number of sick was "increasing rapidly". Whether the "sick" were included in the official MIA is far from clear. They would not have been accounted for by the other two categories. Jackson's report adds that beyond the 2,500 "sick and wounded" there were also "500 persons [at Savage's] having charge of the patients" and – by the way – that as of that day D.H. Hill had collected "probably 1,000 stragglers". OR Vol. XI, Part 2 at 556. The correspondence to and from Lee starting July 3 repeatedly refers to "sick and wounded" in the OR, Series II Vol. IV.
We have no idea when Winder's calculation was performed or how it was determined/collected. The date of the document in the OR, Series II, Vol. IV at 821, is July 16 but that doesn't mean that the number was ascertained that morning or even the day before. On July 16 the Richmond Dispatch reported that a large number were brought to Richmond literally the day before, July 15. Moreover, as the Richmond Enquirer reported on July 14, "[c]asual prisoners are brought in every day".
According to the Richmond Enquirer on May 12, the total number of Union POWs in Richmond on May 11 was 918 and of that number 860 "privates" were paroled and sent off for exchange. (This number closely corresponds to the "886" referred to in the OR, Series II, Vol. III at 553). The Richmond Dispatch further reported on May 15 that "all of the Yankee officers" in Richmond were to be transferred south to Salisbury NC. Regarding Seven Pines, the Richmond Dispatch reported on June 2 that 500 Union troops captured in that battle had arrived as of 5 PM the previous evening. On June 4 the Dispatch reported that "the Yankee prisoners in confinement here were sent South yesterday. About 560 went off." (Once the Union army figured out its casualties the total MIA for Seven Pines was only 647. OR Vol. XI Part 1 at 762). In other words, the evidence is that Richmond was emptied of virtually all of those captured before May 11 and those captured at Seven Pines, either by exchange or by transfer farther south, so those men cannot have been part of Winder's number.
Finally, there is the report by the Richmond Dispatch on July 19 that the total captured in the Seven Days and brought to Richmond by that date was "at least" 8,000, confirmed by the Richmond Enquirer on July 19, which added "[w]hen the returns shall have been complete, it will be found that our estimate some time ago was very nearly the figure, or upwards of nine thousand." We also have Lee's statement in his January 1863 official report of the campaign that "[m]ore than 10,000 prisoners, …, were captured". OR Vol. XI Part 2 at 497-98.
Tenney looked at other sources/methods, as well, but he has clearly undermined the accuracy of the official "MIA/Captured" number for the Federals. He also makes it clear that his "about 4,000" number is an estimate. It is almost certainly impossible to ever come up with a precise figure. Following the Seven Days prisoners were coming in daily from a variety of locations and the authorities in Richmond were scrambling to house all of them, finding and using a variety of locations, moving groups around, etc. Even if the actual increased number turns out to be 2,000 or 3,000, that significantly closes the gap between the accepted Confederate losses and the "official" losses for the Army of the Potomac.