Botched iPhone Battery Replacement - Home button doesn't work?

This is going to be a long one, so brace yourselves.

A bit of context. I live in India and own an iPhone 6S, and Apple only sells and services their products through third party authorised re-sellers. Multiple companies take up proprietorship and operate such stores.

Anyways, my iPhone has aged by about 2 years since I made purchased it, and the battery was deteriorating. The capacity was around 72% percent hence I decide to opt in on Apple's Battery Replacement program, and gave my phone in for servicing around 2 weeks ago.

However, within the first 3 days of its replacing, the phone's battery health dropped to 92%. Finding this amusing, I went in to my service centre to get it replaced however they denied my request and informed me to keep the phone under observation. They even gave me a generic response, -- could be a motherboard problem which I didn't buy into because I never had any such issues prior to the battery replacement.

Fast forward 2 weeks, day by day, my phone started to lag, shutdown unexpectedly, and battery percentages was all over the place. The phone became extremely unstable to the point, the home button wasn't working, phone would just lag constantly, and any time I want to carry out a task on a app, I had to restart the phone, and use the desired app within a small margin of time, else the phone would again reboot and/or hang.

And finally now comes the funny part,

I contacted Apple earlier last night and raised a request stating the same. A rep from Apple India HQ got in touch and instructed me to drop my phone off at any of the authorised service centres regardless of the reseller, hence I decided to drop it off in another brand's Apple service centre, and immediately they noticed the screws adjoining the chassis of the phone and screen (screw ports next to the lightning connector) were never fitted back in after replacement. They advised me to go the service centre where I replaced my battery originally as they had the service records of my phone, and also hold them accountable.

I went into the original service centre, and these guys are in absolute denial about forgetting to bolt in the screws after replacement. After a few minutes, the Apple India HQ rep got in touch and instructed them to look into the phone and they finally agreed to replace the battery, however they informed me the home button is not responsive and they say I need to replace the touch screen since the home button is integrated with this touch screen and isn't sold as a standalone piece.

So as of this post, my home button is works partially. It registers the click when pressed at certain angles, but for the most part it's dysfunctional. Touch ID works completely fine on the other hand.

What I would really like to know is, whether the home button connection could be loose since its partially functional? OR the non fixing of the screws into chassis played a problem leading to this, OR the deteriorating battery play a part in messing its functions?

Any solutions will be appreciated.

Thanks, and sorry for the long post.