> why not charge them more for visas,
Because illegal immigrants do not use visas. So you would be hurting law-abiding people and encouraging circumventing the law - if you can't afford the visa anyway, the question of "should I just through the hoops to get legal visa or try illegal immigration" becomes moot.
> make them pay extra taxes
Why would they pay those taxes? They'd just work on black market and report zero income and apply for welfare. High tax burdens reliably breeds avoidance and lack of compliance, especially in population that is kinda on the sidelines of the law as it is, as it often happens with migrants. And of course it would also sell awfully in the press - imagine an article of how a migrant from Guatemala pays a higher tax rate taxes than a billionaire born in New Jersey - how that would look in next congressional debate?
> restrict their access to welfare benefits
You can do it, but then you'd get exactly what welfare benefits are created to remove - a large population of people who do not have access to basic services, and thus have high level of disease, crime, misery and human suffering. If you accept the idea that welfare benefits are necessary because we should not have too many people like that in our society, then denying people in the country these benefits because they were born in wrong place contradicts this idea. Gating the access to this system on immigration status contradicts the whole premise of having the welfare system. I don't think there's a logically consistent model of welfare state with open borders at all - unless we somehow postulate that welfare spending has a reasonably low upper bound, which I haven't seen proven anywhere.
> Such levies could also be used to regulate the flow of migrants,
Of course you could, but why charging lots of money for visa would be more effective in solving problems we have now than any other method that's been used?
> thus avoiding big, sudden surges.”
Big, sudden surges usually happen due to wars, famines, economic collapses and other such events, in which case I seriously doubt visa prices are going to change anything.