You have a wage, say X. The employer has to pay taxes, insurance and so on for hiring you, based on the wage X. Nothing is removed from the wage at that point. Then you have to pay taxes and insurance based on your wage, that is subtracted from your wage.
Of course this has the effect that the cost of your employer equals your wage, plus employer taxes. And the money in your pocket is your wage, minus employee taxes. Where this is important is when employer taxes change (the employer's cost of hiring you changes, your wage and the money in your pocket don't change), or when employee taxes change (the employer's cost of hiring you is unchanged, your wage is unchanged, but your taxes and the money in your pocket change).
So your real tax burden is the employee's taxes that you pay. What the employer pays is none of your business. When you look for a job, you look for your wages - your employer should know that your wage isn't your cost, but you don't care. In the UK, the employer's tax burden for hiring you can change with your circumstances, for example age, because the government wants to encourage companies to hire people needing more experience. That doesn't affect your wages.
Now if your employer has a budget X to hire someone, they can easily calculate what wage they can offer. They should know that there is extra cost for them. But that's not just the employer's taxes, it's also office space, computer equipment and all that.
PS. There is one situation I know of where a UK employer gets some rebate - I think they don't have to pay the first £4,000 or so of employer NI contributions. With a single employer, they save £4,000 per employee. With ten employees, they save $400 per employee. Now how would you look at that if you claimed all taxes were paid by the employee? The employee would get less money if another employee gets hired?