When someone is dependent on you, denial of that which they depend on you for is a means of interference, yes. A silly example: say I'm dependent on you for air. Normally I breathe just fine, but not if I walk into Room A. If I do that, you refuse air until I leave it. That would be an example of you interfering with my free movement by action of failing to provide the air I require.
If someone is dependent on the government to pay for their healthcare, and their doctors are recommending an abortion, but the government is unwilling to pay for that because they feel the doctor's recommendation isn't justified, then the government is interfering with the healthcare by withholding the money required for it to be provided. It's not a perfect analogy, but it's close. The abortion is Room A here. The money is the air. And healthcare is free movement.
Dependency is fundamentally distinct kind of relationship.
I find your analogies strained and your reasoning rococo.
The government has been not paying for abortions since ... uh, 1776 ... and it appears that
up 'til now this has not been interference with the doctor-patient relationship. But now if the government starts paying to have broken legs set and inflamed appendices removed,
then the act of not paying for abortions suddenly becomes interference? ... this seems a very odd way of looking at it.
You might as well complain that food stamps interfere with people's rights to acquire caviar. After all, the people who use food stamps are presumably dependent on them, and presumably you can't get caviar with food stamps.
By your logic, then, then, not buying people caviar is a way in which the government interferes in people's lives --- so long as they provide them with
any form of food. Whereas if the government gave people no assistance with buying food, then the government would
not be interfering. Of course, the people affected would still not be getting any caviar, but at least they'd be not getting caviar because they weren't buying it for themselves, rather than because the government wasn't buying it for them. And
that is the difference between liberty and oppression.
As for those guys who take away my garbage but won't sweep my yard or clean my house ... don't even get me started.