You don't need to fill all specialist slots all the time (I usually only do it just before my massive GS spam).
If you take 3 policies in patronage you get the policy that gives science from city states which is probably the second best policy for science in the game (after secularism).
Still even in my Babylon game when I hit renaissenance after taking only 1 policy in patronage I wasn't even close to finish rationalism before modern era. I hit modern era at turn 154 in that game which was earlier than the finish of the first vote.
The only way to use the rationalism finisher to enter modern would be to tech slower or possibly by playing Poland.
In my own games:
Science specialists; Always filled
Guild specialists: Also always filled (I don't build the guild until I'm ready to run it)
Engineer specialists: Run when nothing better; happens to cities with the most water tiles first.
Merchant specialists: Also run when nothing better
My capital gets a lot of food via food cargo ships and so is running all tiles worth running before Civil Service and most of the above specialist. Adopting Civil Service causes it to quickly grow to fill the last few specialist spots even with transitioning Food cargo ships into Hammer cargo ships.
Non capital cities: They aren't getting the massive food, but don't have guilds to run either. They tend to start filling Merchant specialists post Civil Service; having already expanded to work all tiles + scientists + some of the engineer slots before then.
Patronage & science: Unless playing Greece, it's a very bad return. City states aren't allowed to build NC (since the early days of Vanilla) and so their science rate sucks. They instead are granted techs that 2 major players have. Even having 5 city state allies is less than the Rationalism opener.
Looks like for you, the policies were the bottle neck instead of science.
Focus slightly more on culture; such as building a guild. But not too much as you don't want to be forced into taking a 3rd policy post Tradition while waiting for Rean.
But the moment that happens, increase focus on guilds and the remaining cultural buildings, but not at the expense of science.
Ensure there is a World's Fair vote and that you get the Silver prize to grant a policy.
Best two filler policies for increasing cultural rate are:
1) If the first group of city states you discover are cultural, then Consulates. Doing the city state requests from a base of 25 (Consultates + Pledge to Protect) will make you friends for a long time
2) If not, then the policies that increase guild spawn rate + cultural building production rate will also improve culture.
Babylon in hands of the human though does have the best science rate in the game (Academy as soon as Writing); so it may be much easier for civs with a bit less of a science bonus to finish Rationalism at the time I stated; but you don't need to be Poland.
