Bike tyre damage from riding on a flat?

So I started riding my 10 year old MTB again and got a flat front tire around 4 miles away from home. Being a novice, I decided to ride it all the way home. Found a hole in the inner tube and patched it up, pumped up and was good to go. Until it punctured again and again and again.

I’ve now got 3 punctures on the inside of the tube (i.e. on the rim tape side) and one of the newest punctures on the outside / top of the tube (i.e. inside the tire wall).

I’ve decided to buy a new tube but really don’t understand what is causing the punctures and think it will happen to my new tube, so want to understand what I should look for. Bearing in mind that my bike is 10 years old and I don’t want to spend a great deal of money as I will eventually buy a new one.

I inspected the tyre itself, it appears to have some damage to it, for example when the tire is pumped to 55 PSI, I can see small cracks in the side wall, these are very few and slight. Also, when the tire is off, the bit of the tire that grips inside the alloy is slightly shredded (i.e. tiny bits of rubber have come off). But this stuff is only visible if you look at the tyre with a close eye thoroughly, your average joe can’t spot these bits of damage so I am curious whether this damage is enough to warrant a tire replacement or enough to be causing these punctures.

I inspected the inside of the tyre to look for debries or anything sharp in there that might be poking the tube but I can’t seem to see anything, the tyre itself pretty much has brand new tread on it and has no holes / punctures.

As the tube is mostly getting punctured on the rim tape side, I looked at the rim itself and it also seems to be fine, however there is places where there is small bits of paint loss on the rim, these places are very slightly to the right of the rim tape so if they are sharp enough then perhaps they can puncture the inner tube? I ran my finger across them and they didn’t feel sharp. It’s hard to tell if they are damaged from me riding on the flat or just imperfections in the paint given the age.

Also, there are punctures on both sides of the tube now so it can’t just be the alloy, my initial theory or hope is that given the tube and tyre are 10 years old now, the tube rubber is just weak and creating holes on its own? Is that possible?

So my question is, what can I do to investigate this further? What products do I need to buy or replace? Inner tube is coming tomorrow.