Rule of thumb, the earlier you pay down your balance, the less interest you will accrue and the faster you will pay off the debt as a whole.
But lets play with some real numbers here. You cited $5000 balance and a $750 payment, but with various bills and things adding onto the balance over the course of a month. Now if your purchases and payments add up to the same number, you are in a losing game, so for the sake of argument I am going to say you are putting $500 + interest on the card each month and making a $750 payment. We also need an interest rate to work with, I am going to use 1%/month and 30 day months to keep the math a bit easier to follow.
You basically have two choices in this scenario, you can pay 750 a month on the card, then use it to make your $500 in purchases/other payments over time as you suggest in your question. Or you can pay $250, and hold back $500 to make those other payments directly without running them through the card, as has been suggested in some other answers.
So let us compare the two...
If I start the cycle at $5000, make a $250 payment on the first day of the cycle, then have no other activity, I will have a balance of $4750 for the month and accrue $47.50 in interest at the close of the cycle. Balance going into the next period is now $4797.50. Carry this out for a year, and your balance at the close of the 12th cycle is $2431.79, and $431.79 of your payments went to interest.
By contrast, if you pay $750 at the start of the month, then add $100 back every 6 days so that you spend $500 over the course of the cycle. You will have an average daily balance of $4466.67, which results in $44.67 in interest charges being accrued at the end of the month. This gives you a balance of $4794.67 going into the next cycle, putting you about $3 ahead of the previous method. Push this pattern out for a year and your ending balance is 2395.86, with 395.86 going to interest. Resulting in a savings of ~$36 over making the smaller payment and paying cash for your other expenses. If this happens to be a rewards card, you also have gained whatever rewards benefit it gives you.
This demonstrates that by the strict numbers game, the scenario you propose should come out a small but measurable distance ahead of making a smaller payment in order to avoid putting things back on the card. So why do so many people adamantly advise you to not do this?
Most of it has to do with psychology and risk. The cash method does not leave any room for you to over spend. You have shredded or locked up the credit card so it can’t be used casually, and when you run out of cash, you can’t spend any more. Which forces you to pay much closer attention to where your money is going.
When you are running things through the credit card, you generally don’t have that hard stop unless you are up against your credit limit, and even then most issuers are quite happy to let you go over and charge you extra fees for doing so. So if you have this plan where you are intending to put $500 on the card in a month, then lose track of something you did early in the month, and inadvertently spend $800, you are digging yourself deeper into the hole instead of climbing your way out.
There is also a risk in terms of income loss. In the cash method, you no longer have the money to spend, and you are forced to make the hard decisions about where to allocate what you do have, making you much more likely to cut back on luxury items to preserve the necessities. In the card method, it is easy to say “eh, the card has room, I can catch up again later” and not realize the mess you are causing yourself until you are in way over your head.
I personally have run all my bills through a credit card in the past so that I could have one single payment to make. Then I was unemployed for six months, and ended up moving before I found a new job. Everything in between, including the move, went on the card. Next thing I know I am carrying a balance of $15k where I used to always have it paid in full. It took roughly 10 years, including several years of working strictly in cash, to get that back under control.
I currently have a card that is carrying a balance, and I am running select expenses (such as fuel and food) through it while I whittle the balance back down. Most of my main bills are still paid directly from cash, specifically so that I don’t fall back into the same trap I did before. Even so, there were several months in the past year where the balance was creeping up instead of down, because we were not paying that close of attention to our spending.
Then my wife lost her job, and it forced us to closely evaluate where our money was going. We still run certain things through said card, but we are much stricter about it being only those select things, and the balance is trending down again. The main reason we are still channeling those expenses that way is because this is a cash back reward card, and we will be getting roughly $1000 back here in a couple more months.