This is the eternal thread on this site.
If "cheap" means badly designed, poorly manufactured and prone to breakage when properly used, no I don't want that tool for general use. If, on the other hand, I am out in the middle of nowhere, and nothing else is available, I would prefer it to having nothing.
If "quality" means well designed, well manufactured tools that do the job, then I would prefer them over "cheap" tools as defined above.
Now, cheap doesn't always mean less expensive. And "quality" doesn't always mean more expensive.
Your post says "on a budget" so that means you, of necessity, must sort out what is what in the marketplace. You will end up with a mixture of less and more expensive tools.
Your post doesn't mention professional use. If that is the criteria, then most pros on this forum lean toward getting the best tools they can find, regardless of price (Even though they also like a good deal). And pros need a wider variety and quantity of tools than the casual user. So to solve that problem, many use the ability to buy on time. But even then, the money conscious ones will only buy the tools they NEED, when they need them. And will also fill in with less expensive tools where they will do the job.
But even for home or casual use, a "cheap" tool, as defined above, just makes the job more difficult. An inexpensive, but appropriate tool makes the job go better. A superb tool, regardless of it's price, is just a joy to use.
Ever ride and maintain an old heavy Schwinn Varsity? Ever get to compare that with a new modern lightweight road bike with high end components? They both get you there, and the old heavy Schwinn will give you more exercise, but which is more of a joy to ride and work on?
So your answer isn't all A or all B. It is the appropriate tool for the job.