Doesn't the last statement contradicts the first one? Are agencies
controlled by the President or insulated from his control?
These statements aren't contradictory. They just reflect a somewhat complex reality (although the new post-Scalia ultra-conservative supermajority on the U.S. Supreme Court has pushed the notion of a "unitary executive", a long time conservative goal, that casts doubt on the constitutional validity of the structures discussed below that previously went unchallenged).
Especially starting around the time of the New Deal (the 1930s), Congress started to vigorously utilize the concept of an "independent agency" such as the National Labor Relations Board, the Securities and Exchange Commission, the U.S. Postal Service Board (formed in the 1970s), Fannie Mae, Freddie Mac, the Tennessee Valley Authority, and myriad other "alphabet agencies." (The Federal Reserve Board was one the first). Congress also experimented with structures like that of the Federal Bureau of Investigation (the FBI) whose director serves for a seven years term, even though the director is a single person rather than a collective board.
Generally speaking, these agencies are run by the Presidentially appointed and U.S. Senate confirmed directors or boards with appointee who serve for fixed terms. As Presidential appointees, they are ultimately controlled by the Presidency as an institution, even if not by the current sitting President. But, since these appointees have fixed terms of office and aren't merely employees at will who may be fired on demand by the President like cabinet secretaries, these appointees have some measure of insulation from direct Presidential direction on a day to day basis in the minutia of decision making.
How much control of The President is enough for a textualist to accept
Chevron doctrine? What if he could hire them but not fire them? What
if he could suggest them what to do but not order them what to do?
The Chevron doctrine really has nothing to do with the amount of authority that the President exercises over independent agencies and the civil service.
Instead, the Chevron doctrine is a separation of powers doctrine that provides that courts should defer to the executive branch (which is ultimately directed by the President, who has near absolute authority over agencies in some cases, and less authority over agencies in other cases), rather than interpreting laws from a blank slate, in cases where a federal agency has adopted an authoritative interpretation of a federal law that is not precluded by a fair reading of the statue in question, even if the court would have interpreted the statute differently in the absence of federal agency input.
The Chevron doctrine tilts the balance between courts and the President in favor of the President, making the President stronger than the President would otherwise have been in the absence of this doctrine. (And, keep in mind that all regulations and litigation of the federal government goes through the Justice Department, which the President does directly control, to a great extent, even when agencies are independent and have significant autonomy.)
The authority of the President to direct tenured civil servants who are not political appointees who can be hired and fired at will, or with U.S. Senate approval, is an entirely different issue with far less of a constitutional dimension until the last couple of years when a conservative supermajority captured the U.S. Supreme Court and has tried to push the non-textual and non-originalist "unitary executive" doctrine.
