How can I have better luck finding a toilet that actually has a seat? (Or an accommodation where the toilets will likely have seats?)
Is it by original design that they were simply omitted? Or do perhaps people have a habit of stealing them? Or something else?
Toilet seat theft is a common and real problem in the Philippines. One of the major reasons listed for the Ninoy Aquino International Airport (NAIA) Terminal 1 being voted the worst in the world to sleep in is toilet seat theft.
http://www.gmanetwork.com/news/story/235691/money/economy/naia-voted-world-s-worst-airport
People list Manila as having the same problem.
http://www.fodors.com/community/asia/avoid-manila-international-airport.cfm
It seems that toilet seats are a complete luxury and theft of the seat and other vital toilet components may be common.
http://an-american-in-the-philippines.blogspot.com/2008/04/bathrooms-in-philippines.html
My advice would be, rather strangely. Carry your own. A variety of products exist. 1 and 2 are in case your fear is falling in (as per @LessPop_MoreFizz comment), 3 is for hygiene.
.I should note I have never been to the Philippines and do not intend to go until they have their toilet seat theft epidemic under control. It has often confused me as to why people think the toilet seat is any cleaner than the top of the toilet without a seat, surely they are equally disgusting. I suspect in all situations (no seat or seat) that this is the best toilet seat.
Unlike the west where people feel a need to cover their toilets, there are plenty of toilets in the Philippines (and the rest of Asia) designed without seats. And plenty more cases where the owner didn't bother with the extra cost of the seat (or replacing it).
Higher end hotels will usually have western set ups with a seat and lid, lower cost guesthouses may go local style. Restaurants that are purely tourist oriented (dinner shows, etc) with again have western style, other restaurants, shops, cafes, gas stations etc will likely be local style.
Is it that by original design, they were simply omitted?
In general no, it's largely because if nobody complains then there's no incentive for somewhere to spend the money/effort to replace a missing seat. In some places you do get purpose made toilets that don't have seats or don't have removable seats however these are obvious and the cases you mention it's likely just poor upkeep but the seats where there originally.
And, yes, people may steal them for whatever reason making them less likely to be replaced.
How can I have better luck finding a toilet that actually has a seat? (Or an accommodation where the toilets will likely have seats?)
Look for places where people would expect seats and complain if they're not there. I.e. higher-end hotels and restaurants, anywhere likely to be populated by tourists or locals with money that will complain about it.
I'd expect seats in any en-suite toilet in a hotel (possibly not in any public toilets they have) unless it's a really cheap place or a remote place that possibly never had them. It sounds like you're implying you're staying somewhere where this is not the case.
Also any public toilet that charges a fee is more likely (but definitely not guaranteed) to be in a better state of repair.
