Airline won't let me pay to upgrade due to purchasing original economy ticket via low-fares site - what's the logic in this?

If it is a private (or “hidden”) fare then the price your travel agent pays for your seat is a commercially private matter between your agent and his liaison at the airline. (It is probably not the fare that you paid him.) It is, to put it succinctly, a secret.

In order to compute the cost of your upgrade, the airline employee needs to know what the difference between the old and new fare is, but that information is not available to him. That applies whether he is selling you up to a public business fare or the business fare on the private tariff. However he is not on the very small list of people entrusted with this highly sensitive information. So in general most employees cannot interfere with private fares.

The logic is that the exact details of the private flight tariff are worth a lot more than the $1000 you were willing to pay for an upgrade. Airlines enter into negotiations with many businesses to sell them private fares, and keeping the prices they offer other businesses under their hats is part of the negotiation position.

It is expected that you would go through your travel agent to reprice the journey, not the airport ticket desk. It may be that the fare is indeed not upgradeable. But a lot of dirt cheap private fares are actually a lot more flexible than the fares on the public tariff: you just have to go back to your agent to sort it out.

It's rather common for unpublished fares to subject to special rules including a "no upgrades" policy. These are discount fares offered through travel agents and consolidators, generally for the leisure market. They offer a way to airlines to sell cheaper space to price-sensitive travelers without having to cut prices for those who may be willing to pay more. Restricting upgrades (and sometimes things like frequent flyer mile accrual, changes, etc...) helps to ensure that these cheap fares stay in their intended market and aren't being used by, say, big-budget business travelers.

For example, LATAM won't give you any frequent flyer miles for unpublished fares (you'll have to check to see whether this applies to your tickets):

Nor shall kilometers be credited for any ticket issued as a LATAM Pass award, nor other free ticket such as promotions or reduced-price or free tickets or free companion tickets, charter tickets, discount tickets for travel agents or industry employees, infant tickets, tickets purchased for items that occupy a seat, unpublished fare tickets, including consolidation rates, or tickets issued subject to special provisions.

And on their page about upgrades (I think this is only about using points for upgrades, the rules for using money may be different) they say:

Only tickets paid in full, on fare profiles Y, B, H, K are eligible for cabin upgrades.

Some tickets purchased from travel agencies are not eligible to use points for cabin upgrades. Check the Sales Center, Fidelidade and Services to find out whether the ticket purchased from a travel agency is eligible for a cabin upgrade.

Simply put, you've bought a rather cheap fare. That's great; nice shopping! They're willing to sell a seat that cheaply because they believe it would otherwise go unsold. Since the marginal cost of carrying an additional passenger is low, while the fixed costs for operating the flight are something they're paying anyway, they'll take your money. But the airline wouldn't be profitable if they sold all the seats at that price, so they can only afford to do this on seats they think might otherwise go empty. As such, they put restrictions on these tickets that make them less attractive, and no upgrades is one of those restrictions.

They would rather sell the upgrade to someone else who is a more loyal (and profitable) customer of the airline, or as Kate Gregory notes, deny you the upgrade in the hope that customers buy more expensive fares in the future.

BA (also One World) is consistently inconsistent about upgrading tickets booked through travel agents. Upgrading with Avios, 30% of the time, I get told I can't as it was 'booked through an agent and I have to contact them' (the agent is often uninterested in helping and/or can't do Avios upgrades). Upgrading with cash, this happens 70% of the time. I've learnt the solution is to say thank you, and ring back and get another agent, for whom this will be no problem. I've even done that and got it to magically automatically update a booking via the travel agent, that then sent me (via some automated system) confirmation of the booking change (most odd). I had lunch yesterday with someone who flies BA a huge amount (Gold card holder for ten years), and he verifies this is par for the course even for Gold card members.

What has always worked for me (subject to availability) is to pay for the upgrade at the airport. This also allows you to skip the checkin line as the agent will check you in / do the passport check / whatever at the same time. These 'promotional' upgrades (according to BA staff) have fewer availability restrictions, and are cheaper than upgrades in advance, as they are apparently not worked out on difference in fare-class, but just different in booking class (i.e. economy to premium economy is the same whatever fare-class economy ticket you had), and they are issued first come first served.

There may be ultra-cheap economy tickets which are not upgradeable at all, but (at least in the BA case) I suspect that may not apply to airport-bought upgrades.

I have no idea whether the above also applies to TAP.