I understand! In your case, I would consider dedicating one room in the house to this, install a split-unit AC and get yourself a dehumidifier as well. Then set the aircon to 65-68F and the dehumidifier to 40%. Allow temps to drop in winter, but limit temp rise in summer. If you notice that RH drops below 25% at some point, consider adding humidification, but I can't imagine this would be required for NYC climate. The combined measures should be more than adequate for home storage.
While lower temps might be nice for the CN's and the slides, I doubt it's going to be feasible to install something at home that will do around 40F and low RH year-round.
If you want/need to do better, I'd honestly initially not try to DIY it but instead look for a museum that can manage the collection with proper storage.
I don't think you can do a whole lot better in terms of archival storage for $5k; there are evidently climate control systems for sensitive collections etc. but I expect you'd be looking at $250k upwards for a setup and that will likely be overkill already for your storage needs.
Also, factor in energy cost of the system over the expected storage period. The few $k of equipment investment is likely to be of a smaller order of magnitude than the operating cost. However, since the latter is spread out in time, it may not be as painful to bear.