(This answer refers to the US investment landscape)
I'm not sure your classification of funds as direct and regular accurately reflects the nature of the mutual fund industry. It's not the funds themselves that are "direct" or "regular." Rather it's the way an investor chooses to invest in them. If you make the investment yourself through your brokerage account, you may say it's a direct investment. If you pay a financial advisor to do this for you, it's "regular." For a given fund, you could make the investment yourself or you could use an advisor.
Note that many funds have various share classes. Share classes may be accessed in different ways. The institutional class may be accessible through your 401(k) or perhaps not even there, for example. The premium class may require a certain minimum investment. Some classes will have a front-end-load or back-end-load. Each of these will have a different expense ratio and fees even though the money ends up in the same portfolio. These expenses are, by law, publicly available in the prospectus and in numerous other places. Share classes with higher fees will earn less each year after fees, just as you suggest. Your intuition is correct on this point.
Now, there is one fee to be aware of that funds either have or do not have. That's a 12b-1 fee. This fee is a kickback to financial advisors who funnel your money into their fund. If you use a financial advisor, he or she will likely put your money into these funds because they have a financial incentive to do so. That way they get paid twice: once by you and once by the mutual fund. It has been robustly shown in the finance academic literature that funds without this fee dominate (are better in some ways and in no ways worse than) funds with this fee.
I suppose you could say that funds and share classes with a 12b-1 fee were designed for "regular" investment and those without were designed for "direct" but that doesn't mean you can't invest in a 12b-1 fee fund directly nor that you can't twist your advisor's arm into getting you into a good fund without a 12b-1. Unfortunately, if you have this level of knowledge, then you probably don't need a financial advisor.