Is it just me or does the skill tree for either offense or defense just not work? I can't invest in Feast without investing in Butcher and the same goes for Block and Unyielding. What gives?

I think you didn't quite understand that he wants to be tanky.

What this guide misses for a real tanky plank is to build many CC-effects. Iceborn Gauntlet is definitly a must have after Rod of Ages, both also have a great synergy. It is also way better than Rylai's Crystal Scepter, as that item does not work on your passive at all and on your ult only when it hits, making it pointless. The rest kinda depends on who you play against, but I would suggest Randuin's Omen for most games aswell. The more CC you have, the more usefull you are as tank, otherwise, other champs just ignore you.

I just want to start by saying thank for the feedback! I have gone ahead and changed the Marks and Masteries too. I really want to get Lich Bane in the build, but I'm not sure what I should replace it with since all items in it have lots of AP and some sort of Tank. Any ideas which I should replace?

Also, I do have to disagree with getting Tear rather than Rod of Ages due to the amount of Health you get from it. If you really want to, you could get it instead, but I don't think it fits in as well. I could be wrong though. I'll point it out in the guide.

Greater Mark of Ability Power is an odd choice and should be replaced with Penetration Marks.
I created a Mastery Tree for you because I'm too lazy to point out your issues. Masteries like Oppression are so strong on Gangplank because it synergizes so well with Grog Soaked Blade and Cannon Barrage (Click here!).
Lich Bane is still a bread and butter choice and instead of Rod of Ages you might wanna consider stacking Tear of the Goddess. Your explanations about Liandry's Torment give the impression that's a weak choice but it synergizes well with his ultimate.