Most people who move up the property ladder either
a) have increased their salaries and can therefore borrow more, or
b) have bought a property which needed renovations, managed them well, and therefore added more to the value of the property than house prices have gone up otherwise.
Let's say that your income stays the same, for ease.
You buy property 1. Property 1 costs £150k, you buy it with a 30k deposit, 120k mortgage. The next house up the ladder would have been £200k.
By the time you come to sell, a few years down the line, house prices in the area have risen by 1/3rd. Your house is now worth £200k. Nice. You sell it. Let's say, for ease, that in the time you own it you've reduced the outstanding mortgage by £10k, so you owe £110k. So you repay the mortgage, and you now have £90k as a deposit.
Your income has stayed the same, so you can once again borrow the same as last time, £120k. Add that to your deposit and you've got a max of £210k to spend (note, I've ignored buying and selling costs here for ease but these would eat in to your cash)
The next house up, the one that was £200k, now costs £266k. So it's still out of reach.
You can buy another house for a max of £210 (minus all your costs) - which is basically going to end up being a property worth the same as the one you were already in. There's nothing wrong with doing a sideways step, we did one ourselves recently because we wanted something totally different. But we didn't end up any better off, we ended up with an asset worth about the same as the asset we previously owned!
The money you have "made" on your house doesn't really do you much good unless you downsize or move somewhere cheaper. It helps people be able to move up because they are generally increasing their earnings, generally paying down their mortgage, and they have an asset which is rising at roughly the same rate (all being well) as the next step up which makes it easier than if they haven't bought in the first place. If they didn't own property they would only have "earned" whatever rate of return they were able to get on their deposit.