Although I have no experience with PSE15, I am confident the purchase of a SSD will solve your problem.
I use PSE13, have 48.000 images in my catalog and use face recognition extensively. My OS drive is a 500GB SSD, I use 16GB RAM and my CPU is a I7.
An SSD will solve mainly the slowness in opening the program. I believe it won't change anything for most current tasks in the editor and the organizer because everything is already in RAM with the 16MB.
We can give comparisons by comparing speed for precise tasks on different computers.
I am using now an I3, 3.1 Mz with 8 MB RAM and classical 2 TB drives. Very responsive... but I don't use face recognition, smart tagging nor any automatic background task. I even have Firefox running and using several tasks (music...)
I need more than 30 seconds to open the program, which is not a problem for me.
Also taking time: opening a 32MB Fuji raw file in the editor: about 8 seconds. Only a more powerful processor would help. Browsing raw files in the full screen or the organizer (F11) is acceptable since next photos are processed in advance.
What is striking in complaints about slowness in Elements or Lightroom is the fact that most are coming from high end computers.
Common problems with graphic cards : practically not used in elements; better to disable them unless you have 4k or 5k high resolution display.
I too had to wait long time for opening the PSE13 catalog while using regular HDDs.
I bought a new motherboard, better and faster CPUs and better graphic cards and more RAM. Nothing much changed until I had the complete catalog on a SSD!
I find every version of PSE somewhat slow to open on most standard hard drive computers and the same for Lightroom. A part of the problem could be PSE Edit and PSE Organiser are virtually two separate programs folded together as one.
My PC was custom built, it is really a Gamming machine, but not used for that its the speed and high level graphics I wanted. All my programmes run from a SSD drive. No data only the OS and Programmes are stored here. PSE on my machine is quick to start but still not quite as quick as other programs, but it's much quicker to start than on my wife's laptop,
As I store Data on a conventional drive I find it much quicker to open PSE the drag the photos in rather than Right clicking on a photo and using "Open With"
How many files in your library?
- I always keep both the organizer and editor open at the same time.
- I have used the organizer preferences not to use any automatic or background task: face recognition, watched folders, file reconnection...
- in the editor, I have used the default preferences, especially for scratch disks.
- If the thumbnails are not fully created, I keep the computer open at night until the spinning wheel in the left of the bottom bar disappears.
Other question : have you many raw files or videos?
You say your not a professional photographer, most users of PSE aren't, so don't let that be a concern. You sound like a lot of people I deal with, in that their interest in photography is based on recording and capturing memories, whether it is of a trip, holiday or some other special interest or hobby.
You wonder why you bought PSE, well if for no other reason it is an exceptionally good organiser that will, if used properly allow complex tracking and bringing together of photos taken years apart or those of a common subject or theme.
You can't see the need for an SSD drive, fair comment, but the time will come when it's time to buy a new computer, do it then. Buying the right spec at time of purchase can extend the life and useability you will get from your computer.