Will airlines object to me booking separate "legs" of the same journey individually?

How exactly would they "not allow" this? What do you imagine they could legally do to prevent it?

However, it usually has severe disadvantages for you:

• In most cases it will be more expensive (assuming both legs are the same airline). It's VERY rare for two separate tickets to cost less than a combined one; in fact in many cases the combined ticket will be cheaper than than the longer leg on its own.
• It may not be possible to check through your baggage, and then you'll have to collect it and check it in again, which can take a lot of time
• If you miss the second flight because of that or because the first one is delayed, you will have to pay for another ticket

I did it once many years ago with a low cost airline that only sells point-to-point tickets (like Southwest but in Europe). As others already mentioned, the downside is that you cannot expect any help from them if you miss the connection (i.e. you might lose the money you spent for the second leg and have to buy another ticket at a premium on the spot or at least pay some extra fee to change the ticket; in the worse case, the return journey for the second leg might be invalidated as well).

Since I also had to collect my luggage and drop it off, go through a passport and security check again (low-cost carriers typically don't offer any connection), it was all really tight even though I had several hours between the flights. That's the reason why you might come across language that strongly discourages it but beyond that, I don't think airlines really care.

Note that some airlines might require you to claim your luggage as a matter of policy but strictly speaking it's not always necessary even if you don't have a single booking for the whole trip, see Can I through check my bags on separate bookings (international flights)? With a bit of luck it could all go very smoothly – but personally, I still wouldn't risk it.

Another issue that hasn't been mentioned yet is that if you want to benefit from a transit exemption in a country for which you would otherwise need a visa, it might be impossible to collect your luggage and remain airside without the airline's assistance (that's not relevant in the US but that's something to keep in mind for other people considering a similar trip elsewhere). Worse case here is that you could be refused boarding on the first leg because you don't have a visa for the destination (that's what happened to the person who posted the earlier question).

From the airline’s point of view:

1. They are selling you a ticket SFO–New York
2. They are selling you another ticket New York–Germany

The first ticket obliges them to get you to New York from San Francisco and back. The second ticket obliges them to get you to Germany from New York and back. Both tickets come with a significant caveat: you have to present yourself at check-in/the gate on time for each individual leg on each ticket.

From your point of view:

1. You get a ticket SFO–Germany, but
2. You need to make sure your bags (if any) are transferred to the transatlantic flight in New York (usually exit the security area, pick up and recheck)
3. You need to be in New York with ample time to do that (an issue for both directions)
4. If you miss the connection due to a flight being late you are counted as a no-show on the second
   • If you no-show on the New York–Germany leg, you lose the Germany–New York leg (but you can have a holiday in New York instead)

5. You will extremely likely need to have a longer connection time at New York, your duty free goods bought in Germany will have to make it into your luggage in New York and your entire trip may be more expensive; give or take a few minor inconveniences.

   (For the record, the return trip doesn’t make that much of a difference since you would need to clear customs and immigration in New York anyway.)

The airline probably would have gotten money for the New York–Germany leg this way or that. As you are in San Francisco, they are now happy that they can get additional money from you for SFO–New York – money that could otherwise have gone to a competitor, low cost airline, petrol stations or Amtrack, in theory. Furthermore, they have little to no additional risk because the missed-connection risk is entirely yours. I struggle to find a reason why they would disallow it, even if the prices are strongly in your favour.

I'm looking at a flight from Salt Lake City, Utah to Sydney, Australia. On Delta, it costs $1215 for a round trip flight that goes through LAX. However, the same round-trip flight to LAX is $168, and the same roundtrip flight from LAX to SYD is $718. So booking the exact same flights, on the same airline, costs $886 instead of $1215. I have no idea why Delta charges so much for the international round trip from SLC, but this is very commonly the case.

In this case, I'll book them separately to save money, but if my connection is messed up, it's the same airline, so there's a decent chance they'll take care of me.

When I've done this on separate airlines, I've made sure that there's plenty of time between my first flight and the connection so that if something goes wrong on the first flight, there's time to catch another connection (even if I have to go buy it from someone else) in order to not miss the big, expensive flight.