Your recollection is pretty good, for the most part. A little help, though:
McClellin's significant history of concussions seems likely to keep him off the roster, one way or another. Net savings = $1.79M.
Solder is currently UFA, so no savings, only an expense if he re-signs. His 2017 cap hit was just over $11M.
An extension-restructuring of Dwayne Allen could reduce his 2018 cap hit much closer to $2M, saving more like $3M.
.
Gillislee's 2018 cap hit is $2.18M. That makes him the 23rd-highest-compensated RB in the league. The net savings from cutting him is about $1.6M.
If he's cut now for the sake of cap savings, and the Pats can't afford to compete with the offers that Lewis and Burkhead get and, thus, can't re-sign them, that leaves the Pats with only White and Bolden at RB, and a gaping hole at the position that would need to be filled with RBs who may (or my not) be as effective as Gillislee was in 2017, who may (or may not) be as cheap as Gillislee, who may (or may not) having any NFL experience, and who would need to learn the offense from scratch.
Cooks' 2018 cap hit is $8.459M (his 5th-year option amount).
If the Pats extend him this season, it could reduce his 2018 cap hit by as much as $4.5M. Such an extension now (or before July, at least) might retain him for another 4-5 years at an APY of under $13M. If they wait until he's a free agent in 2019, it may require something in the $15M+ APY range to retain him.
The net cap savings from Harris' retirement was approximately $1.57M.
The net cap savings from cutting Britt would only be about $895k. Somewhat similar to Gillislee, cutting Britt would be cutting a skill position player with extensive NFL experience and at least some experience with the Pats' offensive system - for negligible cap savings.
The cuts/restructurings you propose, including extending Cooks, might increase the the Pats' cap space to around $37M. However, that still leaves Solder, Amendola, Lewis, Burkhead, both backup OT, and Marquis Flowers, plus core special teamers Bademosi, Ebner, Slater, and King unsigned. Re-signing all of them is likely to hit the 2018 cap for about $30M, with Solder getting about 1/3 of that money. Additionally, the Pats may need to come up with as much as $5M to cover the 2018 cap hits of their draft picks and UDFAs.