Toshogu Shrine

Pros:

• Provides 20 pop space, which saves you 250 wood/500 villager seconds that would need to be spent on shrines for pop space if doing a rush, letting you transition all of your vils to other resources a bit faster.
• Generates 0.35/0.91(0.525/1.085 with kami) resources per second with no/max hunts, +0.033 per shrine (up to 0.66? not sure if it's 1/3 or actually 33%), for a total of up to ~1.745 (possibly ~1.751) resources per second - or about 3.5 unupgraded vils on wood.
• 200 export on age up - Helps get that 400 cost crossbow shipment or 3 fish from the Portuguese consulate if needed, or a military rickshaw/units from the Japanese consulate right away. It's roughly equivalent to 260-280 resources; potentially more with some of the upgrades, but is limited to specific uses.

Cons:

• Requires a large investment in shrines early to reach its maximum resource generation (which delays/slows down unit production), and does not generate resources very quickly.
• Maps often do not have enough hunts for you to both have Toshogu and 20 shrines on hunts, especially with the opponent killing them, so it's usually appropriate to reduce the calculated resource generation by 0.56 (to ~1.016/1.191) putting it at ~2-2.4 unupgraded vils on wood at the shrine cap.
• Extra 33% base rate resource generation from Shrines does not affect XP generation with the 7 lucky gods card, which significantly reducing the value of Toshogu Shrine in many treaty builds which use the card to get as many shipments in as possible within the treaty period (in a 40 minute treaty match, you can ship every card before the treaty ends).

Torii Gates

Pros:

• Military ricksaw saves 200 wood/400 villager seconds that would otherwise need to be spent on getting up a barracks/stable, allowing that 200 wood to be put directly into more shrines or a consulate, and builds the Barracks faster than a villager would (and saving an additional 30 villager seconds).
• A Samurai, which can put some pressure by sieging houses, harassing viles/the enemy explorer, etc...
• Assuming constant villager (14.4XP/min) and 5x ashi production (~65.45XP/Min), and the passive/base 120/min XP gain (72XP/min, the same as a trading post), Torii gates generates 151.85 XP/min. In a sense, it can be thought of as generating 6XP for every 100 resources spent/killed, doubled for buildings/outlaws, so has increasing value over time as your economy improves and you gather/spend more resources. You can roughly equate that to villagers/shrines generating 6%-12% of their resource collection rate as XP - or 2.4-4.8/min for food, 1.5-3/min for wood, or 2.16-4.32/min for coin, increasing with upgrades for villagers, and ~0.36 - 3.096xp/min for shrines.

Cons:

• Value of Experience decreases as more shipments are sent/collected, with every shipment costing 15% more than the previous one (up to 2500XP per shipment at the 17th).
• Experience does not directly translate into resources, but rather units/upgrades, which come in batches rather than consistently over a long period of time, making its value less consistent. This makes balancing your eco a little bit more difficult, as you get a bit less wood income from shrines, and so have to collect more with vils, or ship it.
• Provides less immediate value than Toshogu, as EXP can only be used once thresholds are reached, rather than generating consistent resources, and you usually have nearly 2 shipments banked on reaching age 2.

The first shipment arrives after accumulating 300XP, with each subsequent shipment increasing by 15%(exponential), to a maximum of 2500XP, which is reached after 17 shipments have been sent.

The 6th, 10th, and 13th shipments are when experience can be almost directly translated to resource value for age 2/3/4, as shipments are worth ~600/1000/1600 resources in those ages. Prior to the 6th shipment, experience has more value than resources. The net result is that the the effective value of XP is roughly equal to other resources at the 10th/16th/20th shipments in the best case scenario, though realistically less for age 3/4 shipments, as you'll likely send 6+ shipments in age 2, and you won't have 19 age 4 shipments to send.

Given that 151.85 XP per minute number mentioned earlier for fairly early age 2 if playing aggressively, compared with the ~60.96/71.46 + ~6/7xp (ignoring hunt) or ~94.56/105.06 + ~9.5/10.5XP (Taking hunt on the Toshogu into account) when at 20 shrines without/with Kami.

Torii Gate's resource generation can be valued at ~50% - 150% more than Toshogu Shrine's for the early game (at the ~10th shipment, at which point, the value of Exp declines such that it is lower than resources in age 2.), ignoring any units being traded/killed and buildings constructed/destroyed which increase the relative value of Torii gates.

Even ignoring the DE buff to Torii Gates to 60% from 50% in RE, it'd still be a better eco boost in both the early game and very late game (with, for example, the infinite 6 nagis shipment), with the mid-game value being a bit muddied as shipments become less valuable, while your eco is still at around half of its potential.

Toshogu only becomes better when you either have no more shipments to send, or when XP is worth less than ~0.7 of another resource if you're only making constant vils off of 1 TC and constant 5x ashis out of 1 barracks. Remember that it scales with eco and with your opponent's eco if you're fighting/trading, so can very easily generate hundreds of exp/minute later in the game. With constant 10 ashi production, the Exp increases from 151.85/min to ~217.3, etc... With the infinite nagi/samurai shipment late game (1200 600f + 600c), the worst case is that XP is worth ~0.24f + 0.24 coin. Even if Torii gates is only generating ~ 220XP/min (that constant 10 ashi and 1 villager production), that's still technically better than Toshogu if you aren't banking shipments (Which can be a reason to get the 125 export Spanish consulate upgrade to make shipments arrive twice as fast in treaty/very long games).

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TL;DR: Toshogu Shrine is worth ~1.5-3.5 unupgraded villagers with 20 shrines. 200 export is generally worth less than 300 resources.

Torii gates is more complicated, but:

• It's worth a minimum of ~5 villagers at the point when you send your 10th shipment if you are still in age 2, likely more.
• Worth more before then (close to ~7 or 8 villagers in early age 2 just off of production), and less after, if your eco hasn't grown enough.
• It scales with both your eco and your opponent's (through trading/killing units), and makes trading units evenly in terms of resources a favorable exchange, as you get more XP out of it.

Contrary to what seems to be conventional wisdom, Torii Gates seems like it should be the better choice for the vast majority of strategies regardless of game mode based on the numbers, as the value of being a shipment ahead (ex: an extra 4 vils) is worth more than the income generated by Toshogu, and it can very easily put you multiple shipments ahead.