Is FC timing the best for large mid-castle age push?

First off, I understand the Castle Age power spike of using knights/x-bows to clean up feudal army. Also the concept of FC to start booming asap.

My question here is for something between early castle aggression (or even all-in) and booming. Specifically thinking of closed map like Black Forest or Arena with a large push around 28-30 min. (maybe that is considered late castle if 16 min is FC time - open to discuss best timing)

Why?

Basically if they are completely booming to try and catch them at the time where they have invested in TCs and Vils but not much in production buildings and military. Secretly massing your army hoping they just think you are booming, too, but then showing up with a large enough mass to do damage before they can produce. Of course knights/xbows plus siege would be common but when testing I've also liked Malay with 25 elephants backed by a handful of xbows and monks - or Slavs with 20 something scorpions, few mangonels for TCs, pikemen in front, and handful of redemption monks.

Not claiming this is an amazing strategy - just something I'm trying. Note I'm ~1000 ELO RM & ~1300 TM.

TCs

With the goal to be ready at 28 min I'd assume this is a 1 TC push. I'm not sure how long it takes for a TC to repay itself (resources + vil build time + vil production food) but I'd think 10-12 minutes would be more around break even?

Up Time

The common advice is to age up when you are getting something out of it.

So if you can FC around 16 min but are not building more TCs and probably not having the resources to go full military production yet, it seems like you wouldn't need to go castle age ASAP?

I often find myself pressed for wood for TCs upon aging up when FCing. Though I'm not building TCs here I'm trying to build several production buildings and possibly more farms. If that is the case it seems like I should be staying in Feudal longer and getting double bit axe ASAP? Maybe horse collar, too? (instead of getting them while aging to castle)

But if so, should I be trying to get to Feudal ASAP to get the eco upgrades? Maybe 22-23 vils instead of 25-27 for FC? Or more like 25-27 to age to Feudal then get upgrades and go to castle 30-32?

Math

Sorry for long post - I've actually been thinking about this for like a month - and I know it is all math. I've been tempted to do some math to try and figure it out myself but I'm sure I'd miss something as things seem to domino effect.

For example, Double bit axe is pretty easy to calculate how long it would take to pay for itself - Vils * increased output. But when going FC you usually don't have enough food for it until after you click up. If you delay clicking up - domino effect - there is so much more that it would affect. And when I do this my Feudal seems to drag unless I idle my TC.

I'm curious how long a TC takes to pay itself off, like I said above. Wood and stone to create it, time loss building it as vils aren't collecting, and the extra food to produce each vil - until those extra vils collect enough to break even.

As if you are prepping for the 28 min mark could it be better to go FC at 16, build a 2nd TC and focus on Vil production until like 22 min, then build 3+ military buildings and idle TCs as needed to produce military? That vs one production building ASAP and producing steadily out of one . . . or any other combination in between . . . am I overcomplicating it?

I just play for fun but enjoy nerding out like this lol