According to the rather terse explanation by AP:
The checks seek to ensure that outbound cargo ships carry only grain, fertilizer or food and not any other commodities, and that inbound ships are not carrying weapons.
Pre-war Ukraine was also a major exporter of metals and related products. I don't know if they'd still be interested in exporting those in the context of a war economy, but I'd venture a guess the prohibition on other exports might be related to that. According to the WSJ, prices on the world markets for some of those products went up in the 2nd quarter. Russia is also a major exporter of some of those... and so is Turkey. Before the war, the top export from Ukraine to Turkey appears to have been "semi-finished iron", which I'm guessing got further processed in Turkey in more finished products. (Turkey was also Ukraine's main export destination for scrap metal.)
According to an article from the end of May, despite significant production capacity losses due to the war, Ukraine was still interested in exporting some metal products, but those ran into the same difficulties as exporting food.
FWTW, the relevant bits from the agreement are:
The Parties to this Initiative are the Republic of Türkiye, the Russian Federation and Ukraine proposed by the Secretary-General of the United Nations. [...]
The purpose of this Initiative is to facilitate the safe navigation for the export of grain and related foodstuffs and fertilizers, including ammonia from the Ports of Odesa, Chernomorsk and Yuzhny ("Ukrainian ports"). [...]
Inspection teams will be set up in Türkiye. The inspection teams in Türkiye will consist of representatives from all Parties and the UN. [...] The primary responsibility of the Inspection Teams will be to check for the absence of unauthorised cargoes and personnel on board vessels inbound to or outbound from the Ukrainian ports.
It does not further define "unauthorised cargoes", but presumably they exclude anything other than "grain and related foodstuffs and fertilizers, including ammonia", which make the object of the agreement. OTOH the related UN page is more explicit that
The new Centre will not:
- Facilitate the export of food from countries other than Ukraine; and
- Facilitate exports of containers and non-food items not included under the provisions outlined in the Initiative.
An agreement was also reached with the Russian Federation on the scope of engagement of the United Nations to facilitate the unimpeded exports to world markets of Russian food and fertilizer – including the raw materials required to produce fertilizers.
Regarding the last bit, commentary in PassBlue says:
As for the special arrangement between the UN and Russia to get its fertilizers and food products into world trading, it remains to be seen whether it will work as well. The MoU never says where and how the Russian goods will be exported and puts the onus on the UN to overcome a range of obstacles. [...]
It also implies that the UN will act on Russia’s behalf, saying, “The Russian Federation will inform the Secretariat of the impediments to access of food and fertilizers, including the raw materials required to produce fertilizers (including ammonia), originating from the Russian Federation to the world markets.” And the UN secretariat “will endeavor to engage relevant authorities and the private sector to effectively exempt food and fertilizers” originating in Russia from measures imposed on the country, “based on the principle that those measures do not apply to food and fertilizers.”
Based on that, I'd venture a guess that if other Ukrainian exports were to be allowed, Russia would demand a kind of equivalent deal/treatment, in terms of those corresponding Russian exports being facilitated by the UN, under terms like the above.
Basically, reading a bit between the lines, by the structure of those agreements, Russia has "reflected back" the exceptions that Western sanctions allow for Russian exports, i.e. only the narrow kinds of products that Western sanctions make exceptions for (foodstuffs and fertilizers) will Russia allow Ukraine to export via the Black Sea.