In general, when traveling, most hotels are around ~100-150 a night if not more (especially in larger cities). In places like Japan and Singapore, there are pod hotels meant for single travelers/those on a budget. Is there such an equivalent in Western nations like the USA, even in larger cities like NYC? The closest I could find were hostels, which definitely seem cheaper than a standard hotel room, but I'm worried about not having as much privacy. I've also seen hotel rooms with slightly reduced space, but seem to be just as expensive as a standard hotel room. Are there any other alternatives that I may not be aware about?
There are not a lot of options equivalent to pod hotels, though some concepts in the same family do exist.
Some hostels have private rooms (sometimes still with a shared bathroom), which may be a good option those who want a hostel but with more privacy. These vary significantly in security and design, so you'd have to research that.
There are some micro-style hotels in expensive cities in the United States and other countries, though. Yotel has rooms starting around 100 sqft—not pods but quite compact. The Pod Hotels in New York City are a similar concept, and there are a few others in this category. While these hotels are cheaper than their larger competitors, they tend to be located in extremely expensive (in non-pandemic times anyway) cities, so rooms may still be over $100/night.
Consider it from the perspective of the hotel developer (pre-pandemic anyway). If you're in a city like NYC where hotels have very high occupancy at high room rates, what reason is there for you to try to strip space and amenities down to the point where, say, $50/night rooms are profitable when you can evidently make a solid profit off much more expensive standard hotel rooms?
I suspect many travelers in these countries that want budget accommodation prefer hostels or AirBnB short-term rental rooms over true capsules (which may be forbidden by "western" building codes and zoning anyway). Once you're to the point where it's a trade-off between a 100 sqft clean and well-maintained micro-hotel room vs a 250 sqft regular hotel room, you're more in the territory of comparing different levels of hotel costs than a notably cheaper category of lodging.
There may be some other forms of low-cost lodging that could be worth considering in some places. In some cases, it could also make sense to stay in cheaper lodging far outside the city center and use public transit to commute in.
While this doesn't expressly address the "are there single traveler, pod-type hotels in the US?", it does offer a solution that the pod-type hotel would - lower cost.
The cost of the hotel room in a particular city (in the US, at least) is dictated a couple of factors:
The "chain" or brand of hotel you're booking.
Hilton is more expensive that Motel 6. Always and forever.
The location of the hotel.
Downtown is more expensive than the suburbs. Generally. Some suburbs have a large business sector and are more upscale than others.
The general concept is that business travelers aren't paying for their own room, so they're less cost sensitive (hey, the company's paying, might as well have a nice place, right?), but they're highly convenience sensitive (I don't have time to travel from the work place to a cheap place in the 'burbs, especially after wining & dining a potential client until late in the evening). Additionally, the business traveler may not have private transportation (a rental/hire car) and be fully dependent on public transport and/or simply walking from the work site to the hotel. (I was in that situation for a new job - working remote, but spent a week onsite to meet people - my hotel was literally across the street from the work site.)
However, individual travelers on vacation/visiting family/etc. are paying for their own room so they're more price sensitive and less convenience sensitive.
If you need to visit NYC, consider staying across the river in NJ. Hotel rooms will be cheaper (even for the same brand) because location, location, location! You'll pay for it, though, in additional travel time, cost and inconvenience in having to get in and out of the city each day you need to make the trip. This is exactly what my wife & I did when we visited my family in Manhattan with our young kids a decade or so ago. We stayed in Jersey and made an adventure of traveling into the city each day so it was fun for them. We took the tunnel, we took different bridges, etc. As a family on vacation, we could do that. As a business traveler, that would have been a nightmare.
Generally speaking, you're going to get the same quality and amenities by choosing a suburb Hilton v a downtown Hilton (substitute Motel 6, if desired), but you're going to pay less the further away from the city center you go.
There are single room options in regular hotels, if not all and often not many per hotel even if they do exist, so likely to be booked out when you book on short notice.
How many and how much cheaper does depend on location and the market they serve.
I find them when using a hotel booking site which allows searching for a single occupant.
Another example of pod hotels meant for single travelers/those on a budget: https://www.pandapodhotels.com/ (Vancouver, Canada), where I happened to stay recently
In Western Europe, you will find discount hotel chains like F1, Première Classe, B&B Hotels, or Ibis Budget. They do not offer capsules but rather rooms for up to 2-3 people, often en-suite. They seem to cater to a similar market, though, and easily beat your $100-150 price (typically under €50 for a room, even for single occupancy). They are common in midsize towns but they exist in major cities, too. They are usually located outside the city center and easier to access by car than public transportation.
Possibly not the geography you're looking for, but there is a pod hotel at the departures area of Moscow Sheremetyevo airport. It has about a dozen pods, maybe a bit more. I don't remember the terminal, though - it was an Aeroflot flight code-shared with LOT to Warsaw in June this year, if that helps. I also don't know the price, or about other airports - I didn't see any at Vnukovo, but there I only saw the arrivals area.
In any case, this is evidence that the thing you're looking for is findable outside of Japan.
Yes, many hostels in Europe have been switching over to pod hotels in recent years. There's a pretty nice one that I've seen in Tallinn, Estonia. Just search "pod" or "capsule" on a booking site and you're bound to find a few. I do not think this trend would apply in the USA, where hostels never really took off in the first place.