That's already blocking off a decent % of the day day, meaning the effective visitor capacity is similarly reduced.
Not to mention that having to force everyone out at scheduled times. When I was there we visited several mosques (also amazing places) and our guide scheduled our visits around prayer times and it made the logistics much more complex for those visits.
I also hate invoking the slippery slope arguments, but, if they were going to restrict access entirely (as with some mosques like Al-Aqsa and Dome of the Rock in Israel where I was last year) they would start by simply restricting things a bit. It's hard to see.
All that said, Turkey is free to do what they want and the Turkish people are going to support Erdogan. I'm just sad that in the future other people may never get to see this incredible site.
The Blue Mosque just across Hagia Sofia Mosque also allow visitors even during the prayers, except probably during the Friday prayer since it is really crowded. I never had the change to attend Friday prayer there so I am not sure, but I did at other times. Tourist are allowed to visit during prayers and even has the opportunity watching the prayer.
I think it will be very similar if not exactly the same manner. No worries about the visitors of any religion or of no religion. You can experience these majestic structures. I recommend Suleymaniye Mosque, which is much larger than then Blue Mosque but not as big as Hagia Sofia. The mosque at Edirne is from the same architect (Sinan [1]) and bigger than Hagia Sofia.
Current Hagia Sofia is actually the 2rd rebuilt of the original structure. Original was like the churches in Europe and more rectangular than this one, and did not have a dome. The last structure had dome initially but collapse couple of times and was rebuilt. The reason that it collapsed each time is due to the rectangular shape of the base and did not have enough support of half and quarter domes around it. Sinan the architect is actually the one succeeded to keep the dome from collapsing.
There's also an ayasofya in trebizond that was converted back to a mosque in 2013. That building is still open for visitors but unfortunately the main dome and its mosaics are now obscured by barriers put up during the conversion.
It would be a trajedy if the same thing happens in Istanbul. Hopefully it's enough of a tourist attraction that it doesn't happen.
Ayasofya is an incredibly impressive and beautiful building. There's nothing else like it in the world (even though it was a major influence on subsequent buildings for a thousand years.) Far more beautiful than European churches.