Is it OK to have a PhD thesis with shortcomings and inaccuracies? I recently defended my PhD thesis and was awarded a pass with some minor corrections. I am due to submit the final version of my thesis very soon.

The examiners were both very happy with my thesis, which explains the final verdict. However, as the saying goes, no one understands a thesis better than its author, as the person was invested in it for years. When polishing the papers to send for publication, my supervisor and I found a major issue with the thesis, which although it does not invalidate the overall findings, it would change a significant portion of the methodology in my thesis. While I am currently addressing these issues for the publication of one of the chapters, as well as my job market paper, this issue has to led to severe depression and has made me feel inadequate as a researcher. I worked incredibly hard and had a particularly difficult PhD process; so it is difficult for me to find that at this stage (post defense) I am submitting a thesis that is predicated on a methodology that should have been approached entirely differently; and frankly while in my corrections, I have in essence remedied the problem so that it fits my analysis, changing it as I had intended it to would imply changing a substantial portion of the thesis. This in turn has caused a feeling of despair and inadequacy.

I spoke to a friend who is currently working as an associate professor and he comforted me by saying that irreproducibility, inaccuracies, and mistakes, be it major or minor are very common in vast majority of PhD theses and that I should not beat myself up for this and that the said PhD title is well deserved. However, I thought I ask for some second opinion.
The thesis is a "good" one if you have passed and will be awarded your degree. Don't overthink it. You have learned something from producing it that you can leverage into future work. That is, in lots of ways, a big advantage. If your advisor is also happy and wants to work with you on any future extension, you have a positive outcome, if not a perfect one. 

That the thesis "might have been better, if only..." is true pretty frequently. 

My suggestion is to take the advice of the reviewers and get done. Then write a future paper extending and improving on the thesis. Don't make it harder than it needs to be. Your dissertation shouldn't be thought of as your life's best work, only its first. 

Don't let the perfect be the enemy of the good. (apologies to Voltaire)