While waiting for other and possibly more experienced advice, may I offer a few comments. In the first instance (oh and of course welcome on board

), if you wish to try to match the bleached out wood to the darker one, use of stains either directly or included in the re-finishing product can help. The problem being though which stain to use and/or at what strength, how to apply etc.
From my experience, this is possibly a trial and error exercise. If so and especially as you want to get it right, it could be worth trialing on some scrap timber to see how best to achieve the desired result. Most hardware shops etc carry a range of stains.
Re the piece you have stripped back, I would have to say to me it most definitely looks like pine, not oak, as the grains for each timber from my experience are totally different. Happy to be advised otherwise.
The Howards products are supposed to be excellent, though I don't have any experience with them. Ditto the Ubeaut products advertised through this site/forum moderator.
What finish to use, to restore, that I guess depends on what you wish to achieve with, durability you are seeking etc. At a basic level french polish (shellac based finish) is the most typical for antiques, but it is vulnerable to marking due to water, heat, and/or just scratches. It is easy to repair though, and when done well, complete with a bees wax polish IMHO is superb. It is what I mostly use. UBeaut also sell a hard shellac finish, which I believe negates some of the problems with heat and water. I intend trying it.
Other finishes though more typically involve oils, such as Tung, Danish or Linseed, or even Organoil. I have used Tung for floors, and wet surfaces (ie. bench tops/cupboard doors) etc. I have also used organoil also for a benchtop, next to a stove to see how it worked. I wax it pretty regularly and it seems fine. Linseed from my experience, is pretty short life and not all that user friendly from a drying perspective. I am sure Danish is good also, and I have known of people to apply it to furniture.
The one product personally I would stay away from is polyeurathane based finishes for old furniture. Good luck with your projects.