Try not to combine color gamut and bit depth.
If you capture in raw, you are getting everything. After you have edited your raw file, you can convert it to ProPHoto gamut, and 16 bit depth - I use PSD since all the post processing software I use reads it. TIFF is just as good. When I am done processing the image, I check for out of gamut colors, then convert to sRGB or Adobe RGB depending on where the image will end up. For print I use AdobeRGB, and for web, social media etc I use sRGB. Rendering intent can also affect how a printer will deal with out of gamut colors. Again, if your image has no out of gamut colors, you are not likely to see a difference between relative colorimetric and perceptual rendering intent.
As far as any differences between the color spaces are concerned, it all has to do with the image you are processing. Most are nearly or completely within the gamut of both the display and the printer. You will not see any differences there. Most printers print a larger gamut than sRGB, and nearly all will print some colors that are not viewable on a display.
This article may shed some light:
https://fstoppers.com/pictures/adobergb-vs-srgb-3167On some images I can see a difference between my 8 ink Epson 4880 and my 12 ink HP Z3200, and others not so much. My prints usually look more vibrant than the display.
I think your mistakes are is making the assumption that if a device claims sRGB (or any other) color space coverage, it is the same as any other device claiming the same, and believing that you need to convert to sRGB to print. There will always be a bias and gamut mismatch (which is why when color is important you need to profile every component in the workflow, including the camera, display and output device (printer, projector, etc) and if using film the scanner, it should also be profiled.
Most photographers don't need that level of control over color. You can produce really great prints with only your display profiled, and using OEM inks and paper in your printer.
I use large gamut 14/16 bit files throughout my workflow, from capture to final, and I select the color space appropriate to the output device at the very end.
Try not to combine color gamut and bit depth. br ... (