Wood stain is not tacky but still leaves residue on cloth

Per the question, I used Minwax until it faded, but it was still quite obvious and visible (close to the last pic in the question post).

Then, taking a page from the "mayo and ashes" tip, I mixed 1-part mayonnaise (Hellman's) to 1-part baking soda (not baking powder; I used Arm & Hammer brand) as a cigarette ashes replacement. I rubbed the mixture along the grain directions using a paper towel and some pressure.

The stain faded, then vanished. The resulting wood is smooth, and considering the baking soda is abrasive, appears unscratched. After wiping off the mayo mix, my table may smell like a turkey sandwich, but it's beautiful again.

Now all I have to do is polish off all the Minwax buildup from my earlier cleaning attempts.

I would never have thought of going with the grain had I not spoken to the lady from Restore and Adore, and had the baking soda idea fell through, I would have gone with the "last resort" 4 0 steel wool with a lot of Minwax applied.

Pic of the repair:

Congrats on your first project! I understand your concern. Stains don't offer wood protection so a top coat is advised especially if it is for an outdoor table.

I used a spray once and was disappointed, so my background is with a can. But I would suggest a foam brush, they are cheap, can be tossed afterward and when applied correctly they don’t leave any brush marks.
If you run smooth and consistent you should not get any bubbles or streaks.

Preparation and environment are important, make sure the surface is clean and sanded smooth.
Also a light is important, so you can reflect light against the table to spot any issues. Poly can be understanding if done right. Take your time and make sure you have enough time to complete the project, and avoid doing it where bugs, wind and small debris can land and then stick to the surface before it dries.

Don’t shake the can, that prevents bubbles within the poly before application. Dip the brush in and run with the grain slowly.
Make sure your application is thin enough, a lot of people want to apply it too thick, which makes the process much more complicated.
Run long straight lines with the grain. When running the next line, make sure the brush over laps the previous run.

Once the project is done, let it sit, don’t poke or try to fix anything. Poly over time will flatten itself out as it dries.
A warm dry day is the best environment.

If you see spots or bubbles wait until it dries sand them with >=280 grit sand paper.

If the stain is water based use a water based poly, or if oil based use an oil based poly. Both have pros and cons. When applying a second coat, sand gently to prevent sanding out too much poly and go smoothly with the grain. I avoid steel wool because it can leave small pieces of metal. Make sure the surface is very clean before the second coat.

Good luck and enjoy the table!