(Note: does everything sound natural? Could you suggest me other sentences that sounds better? )

I'll do my best!

Local markets, where our parents and elder relatives always bought their daily chopping(?), hasve been steadily replaced by large supermarkets.

Do you mean "did their daily shopping"?

These small business hasve been part of our lives both in cities and villages, due to, especially, because they keep a closely contact with customers.

I would add a descriptive word, like "an important part of our lives."

However, the way of life has also changed, pushing ourselves. We are pushed more and more to buy at supermarkets instead, mostly becausedue to lack of time.

Your sentence is a bit too long.

Both options hasve their pros and cons.

At supermarkets, the main advantage is saving time, so w. We can get all kinds of products without having to move from one side to another, although quality, evenand often price, is pretty lower.

Again, this sentence is a bit long. Some words you might use in place of "pretty": somewhat, generally, noticeably, often.

Conversely, the "traditional" way supposed, you have to go to the butchery, the greengrocer, the fish market,.. etc., where it is fouyou can find fresh products, thoughbut you have to go every day or spending a whole morning just for weekly shopping.

All in all, it will depend on my time that I choose an option or another.

Something like "which option I choose depends on how much time I have" might sound more natural.

More often than I would like, I have no choice but to go to the supermarket, it is just a matterjust because of time.

NHowevertheless, whenever I have enough spare time enough, I enjoy buying at different places, looking for the best products in order to cook a special and tasty meal.

"Nevertheless" sounds strange here.

So, after all, it deserves taking all this time.

Maybe something like "It's worth taking the extra time."

Finally, in reply to the essay's question, I prefer the older style, in which remainswhere communication with the client, as well as remains, and because of the product quality's concerning.

However, stressful weekdays unable us to keep this way, and weekends are conceived to rest rather more than going shopping.

Maybe something like:
"However, stressful weekdays make it impossible to always be this way. Usually we prefer to rest on weekends instead of going shopping."