This probably isn't what you want to hear, but:
You seem concerned about getting the chain super clean. That's going to be very hard with it on the bike.
Any dirt on the chainrings/cassette is going to get put back onto the chain as you rotate it.
Any cleaner/degreaser you use may well make its way into the wheel bearings, the derailleur jockey wheel bearings, the derailleur pivots, and perhaps even the bottom bracket bearings. The more cleaner/degreaser you use, or the stronger it is, in an attempt to get the chain "really" clean, the greater the likelihood of damage to bearings. You're trying to degrease the chain, but grease (or oil) is exactly what you want to keep in all the aforementioned bearings.
Again, this probably isn't what you want to hear, but I would learn how to get the chain off. Assuming it's got some sort of quick-link/masterlink then ideally you need two pliers - one opens the quicklink, the other closes it. They seem to be ~£10 each. There are other methods involving string and suchlike (google it) but I couldn't be bothered. Removing and replacing chains is a fairly basic bit of bike maintenance which is well worth knowing. And if you ever break your chain it might well be a lot faster to put a new one on yourself. Or if you put new sprockets/chainrings on and need to change chain length.
Also one way of reducing drivetrain wear is to rotate chains.
Somebody up thread said that quicklinks are only meant to be used once. It may vary by brand, but the KMC 9 speed chains I use come with a quicklink that KMC specifically says can be opened/closed three times.
With the chain off the bike you can soak it in the degreaser of your choice, agitate it, leave it overnight, or whatever. You can get it super clean that way.
I understand your desire not to buy unnecessary kit. I have both quicklink pliers, and one of the "plastic chain cleaning gadgets" that you refer to. I've used the pliers lots. Whereas I've used the chain cleaning gadget exactly once, and will probably never use it again. It was a filthy disgusting mess which resulted in all sorts of gunk over me and the patio. Do yourself a favour, buy pliers, clean the chain in an old jam jar, and forget the "magic cleaning bath".