Solved independently of @El-Guest (kudos for being first to find a solution to the crossword), with two differences:
Here, all words are capital cities of countries that are sovereign states if we use YAOUNDE (capital of Cameroon) in place of @El-Guest's PAPE'ETE (capital of French Polynesia) for 31 across. This then means we have TASHKENT (capital of Uzbekistan) in place of @El-Guest's CASTRIES (capital of St. Lucia) for 24 down as well.
The resulting letters in gold boxes are:
MSUZEDABIBLOPROTACRIMWAEEEAULT - this differs from @El-Guest's list in having a 'T' in place of a 'C'.
Rearranging these letters can produce the following 4 country names:
ZIMBABWE, ECUADOR, TIMOR-LESTE and PALAU.
I'm pretty confident this is a unique combination of capital cities - the lack of an N limits the potential countries quite drastically. Also when consulting a list of all sovereign states and considering only the letters available from the gold boxes, we find the following limitations:
- Need a W: MUST have one of MALAWI or ZIMBABWE;
- Need a D: MUST have one of BARBADOS, CAMBODIA or ECUADOR;
- Need 2 U's: MUST have two of AUSTRALIA, AUSTRIA, BELARUS, CUBA, ECUADOR, PALAU, PERU and RUSSIA.
The only combination that left letters which could be anagrammed to another country of the world involved using ECUADOR for two of those categories, PALAU as the other U-containing country, and ZIMBABWE for the W. The leftover letters spell TIMOR-LESTE.
What is the connection between these 4 countries? Well, there are potentially several - but one lends itself naturally to an object that could fit into the drawer with the postcards, which could well have been:
A US dollar bill, since all of these countries use the US dollar as one of their official currencies.
(Also, note that another pleasing connection with relevance to this puzzle is that none of their capitals appear in the crossword (Ecuador = QUITO, Palau = NGERULMUD or MELEKEOK, Timor-Leste = DILI, and Zimbabwe = HARARE), meaning that Uncle Wendale is yet to visit them!)
A brief note on the solving process:
Start by placing ANDORRA LA VELLA in the leftmost column and GUATEMALA CITY along the bottom (these are the only 14- and 13-letter capital cities of sovereign states). Build out from these by considering letters shared by two capitals and evaluating the possibilities. The grid effectively fills towards the top-right corner.
Finally, you end up needing to choose between (i) using ZAGREB, REYKJAVIK, BEIRUT and DAKAR and (ii) using MANILA, ISLAMABAD, ASMARA/ANKARA and DHAKA. Either would be a valid crossword fill, but (a lot of) trial and error established that only the first option yielded 4 country anagrams for the final solution.