My friend asked for help fixing loose backs on his dining chairs.  I have not seen them in person, but he sent me photos. 

The chairs are solid oak, backs are mounted to the seats with a wedged split tenon and have developed play over the years.  I have a high pressure glue syringe that I've used on my own furniture with good results (drill small well placed hole and force Tite Bond into joint) I plan to use the glue syringe but am wondering if also gently tapping the wedge tighter or even adding a pin nail or two might be helpful.  Concerned that tapping the wedge might have unintended consequences like splitting or forcing the chair back out of the seat.

Any input appreciated, photos attached below.

For what is it worth, this older chair is almost certainly assembled using hide glue.

Hide glue is highly praised by furniture restorers because it is readily reversible.  That is, with the proper application of heat/hot water, the bond can be broken and the parts can be disassembled.

A furniture restorer would probably disassemble the loose joints and then reassemble.

I have not removed hide glue, but there are articles on the process online.  Here is one:

https://www.popularwoodworking.com/techniques/reverse-hide-glues-bond/

And this is my Google search:

https://www.google.com/search?q=removing%20hide%20glue&ie=utf-8&oe=utf-8&client=firefox-b-1-m

Not much advice from me.  But, regarding the suggestion that this is hide glue.  I am not so sure.  It appears these are somewhat newer chairs.  From the 50s, 60s, 70s most likely.  Not antique where hide glue probably would have been used.  They look like the brown oak furniture style of the 1970s.

As for repairing them.  Looks hard to get a clamp on them to drive them tightly into the seat.  And putting a pin, finish nail, from the seat edge might maybe work.  Short distance.  But would the pin do anything?  Without new glue to anchor everything in place, or without a new wedge to secure everything, would the pin actually hold the loose chair back post in place?  Kind of like grabbing a chain in the middle.  Where you are grabbing is secure.  But the loose chain is still moving on both ends.  The pin isn't going to secure the whole post.

Without pulling the chair back posts out of the holes, sanding, cleaning everything, new wedges, and then gluing them back together.  Maybe with epoxy.  I'm not sure you can repair the chair backs.  A repair that will keep them solid for many years to come.  The wedge is in the joint to counteract the problem of the end grain in a round hole.  Gluing long grain to end grain does not work and always leads to expansion breaking a glue joint.  The wedge helps with mechanical clamping.

You can probably use a wood screw in the wedges.  With a small pilot hole and a oversized screw, it would apply a mechanical force pressing outward from the screw.

I’m pretty sure it would tighten the joint, but I’m less sure about the longevity of the repair.

Note:  I believe I saw this in a magazine many years ago.  I have never tried it myself, and I have no idea how effective it would be.

The one thing in its favor:  It’s quick and easy.

The apostrophe with an s has been a pet peeve of mine for years, but I always have to consider the context. In a place where English may not be everyone's primary language, mistakes happen.
English is so overly/needlessly complicated that many, who have spoken it their entire lives, can't do it properly.

On packaging or assembly instructions though, I always wonder, "Why don't they have this reviewed by a native speaker?" So many things do not translate correctly.
Even the "English" between America and the UK don't translate 100%. It may be another English word, but we don't use them the same. When explained or well thought out, it's understandable, but not the same.
In the US we would call a battery with no charge "dead", in the UK it would be "flat".
We would use "flat" to describe a carbonated beverage that has lost its bubbles or a tire that has lost its air. The UK would call the beverage "still" and the tire a "puncture".
They both make sense, but not the common usage.
Tacking on the apostrophe, I just don't get.

I guess the first question to ask is this supposed to be just a temporary fix or does your friend plan on using the chairs for a long period of time. Quick fixes can be easily accomplished with lots of glue, screws and nails.

If this were my project, and your friend is looking for a permanent fix, I'd start by removing the wedge so that the back can be removed. Once the back is removed you can then determine the severity of the condition and the best method to fix the chair.
The 2 holes for the chair back could well be bell-mouthed on the top and bottom surfaces of the chair seat after years of movement. Likewise, the chair back could have grooves or ridges on the tenon mating surface.
Is the chair back tenon diameter too small?
Is the chair seat hole diameter too large?
Are the diameters, truly diameters or are they obround?
Is the chair back tenon cracked/cracking from excessive wedge pressure and years of chair back movement?

That is not really an abbreviation, which is shortening a single word. These are contractions, combining two words and cutting a few letters out.
I agree completely, the would of thing drives me nuts. I think it's because of how the words end up sounding, then people get lazy.
Words with several syllables, especially the ones which are not articulated very well, get the lazy treatment.
Did you ever hear someone say "prolly" when the meant "probably". Difficulty words get butchered or dropped from daily use, so people don't have to feel awkward.

Way off-topic, sorry.

Even the "English" between America and the UK don't translate 100%.

You mean doesn't translate 100%.  I agree.  Or does not translate 100% if you are against using apostrophes.