So for the record, it was 26 calendar days to receive my timer modules from AliExpress (estimated delivery time was 30-50 days).
And having been "burned" multiple times by these low-cost modules from Asia, let's be absolutely clear on what I am talking about. Here's what the modules I received look like. Note the "1843" marking on the right image of the circuit board. Curiously, if you look at the online photos of some of the online listings, this marking is nowhere to be found.
But including shipping, these are essentially $1 DELAY OFF timer modules with a broad operating DC voltage range.
Here's my guess at the schematic. Let me repeat myself. This is just a guess!
The real issue is how the "output" pin and polarity operates in terms of if/how it can be adapted to become a DELAY-ON timer module. As Dusty observed, the behavior of the output pin of what is surely a custom-programmed microcontroller (uC) 14-pin chip is the key. Since the transistor markings cannot be reverse-search to determine actual part numbers, I simply draw them as 3-terminal components. I don't think it relevant to devolve into the nuances of whether these are BJT or FET devices. If you even know what I'm talking about then good for you but again I don't think relevant to the matter at hand!
So here's the "thing." When power is applied to the module there is a brief delay during which the module OUT pin (Output solder blob set to "-" level) is incorrect until the module uC "boots up" and properly sets its pin 1 to ON level. Note that inverting transistor Q2 is between the uC output pin 1 and the module OUT pin; during the startup interval, the OUT pin is pulled to V+ by resistors R3+R6 or about 24K.
I can readily duplicate the brief flash when using the 10-cent 2N3904 inverter circuit previously discussed. Here's a video of this brief flash followed by a 10 second delay then the 5V LED strip turning ON. This interval appears to be about 0.05 sec...a scant 1/20th second but nevertheless enough to effect a "quick flash." The goal is to eliminate the brief flash - hopefully the flash can be seen in the video:
Dusty, please confirm we are on the same page as it were. I'm pretty sure I will be able to come up with an inexpensive modification that converts this to a DELAY-ON module that can drive either 5V or 12V LED strip sections and masks that brief start-up flash.
BTW, let me know what your parts "stash" looks like. Do you have capacitors lying around? Do you have an assortment of resistors lying around (or just 1K resistors)? Etc. I have a rather large parts stash and if all you need are a few tiny parts that are less than 1/4" thick (per USPS rules), I will gladly send them to you with my compliments...or you can choose to send me 75 cents which I believe is the current cost of a non-machinable postage stamp...LOL.