I second Joe's enquiry (or going by upvotes, I third it). What has led you to think the first part is true? Congress has delegated huge swathes of its powers. In J.W. Hampton, Jr. Co. v. United States (1928), SCOTUS held that delegation of legislative authority is an implied power of Congress, provided there is an "intelligible principle" given by Congress (via legislation) to guide the Executive branch. Violations of the "intelligible principle" test are rare, with the last example of a law being struck down due to failing the test occuring in 1935, and concerned the National Industrial Recovery Act.
And that's why we've got all of these executive branch departments, and a bureaucracy that certain political circles decry as bloated and overgrown. The EPA, FDA, and IRS are all examples of well-known Executive branch departments/agencies which exist via a delegation of Congressional authority.
As for a relatively modern take on Congress's ability to delegate, in Mistretta v. United States (1989), SCOTUS stated that:
Applying this "intelligible principle" test to congressional delegations, our jurisprudence has been driven by a practical understanding that in our increasingly complex society, replete with ever changing and more technical problems, Congress simply cannot do its job absent an ability to delegate power under broad general directives. Accordingly, this Court has deemed it "constitutionally sufficient" if Congress clearly delineates the general policy, the public agency which is to apply it, and the boundaries of this delegated authority.
As such the delegation of some legislation authority to DC via the Home Rule Act could be sustained under the "intelligible principle" test. Though insofar as Congressional delegation is normally done exclusively to the Executive branch (via a "public agency"), rather than some piece of non-federal government, the second point you bring up may also be relevant.
For this, the use of "exercise" in the phrase "power...to exercise" is important. Just like you, the government can always choose not to exercise. The clause doesn't prohibit the possibility that other entities can wield legislative authority within the area, it simply says that Congress is the final and ultimate authority on all legislative powers used therein. Indeed, the existence of a local governmental authority in D.C. was conceived of by the founding fathers, with Madison writing in Federalist No. 43 that "a municipal legislature for local purposes, derived from their own suffrages, will of course be allowed them".
The particulars of the Home Rule Act also make it clear that Congress retains ultimate authority. New legislation passed by the local government, including their budget, still must be approved by Congress, and they are prohibited from addressing certain issues. One of the complaints of residents concerns exactly such provisions, as legislation passed by sizeable majorities by the council can still be blocked by Congress. And sometimes Congresspeople will try to hold up or wholly block a piece of local legislation in an effort to gain concessions to their legislative agendas elsewhere, and they will also frequently vote for/against local legislation based solely on the politics of their constituents (who don't live there) and without regard for the city itself. Gun control in DC is an issue that Congresspeople frequently try to regulate through their own politics and constituents rather than the wants and needs of DC itself.
But that's getting us off track. DC has received various degrees of grants of legislative authority prior to the Home Rule Act, and SCOTUS has upheld those. Two examples of many (for others, see e.g. footnote 22 in this article):
- District of Columbia v. John R. Thompson Co. (1953) was a segregation case that pivoted on whether laws prohibiting segregation formally in effect under local legislation passed in compliance with Congress's dictates, had in fact been de facto repealed when Congress revoked that legislative authority and restructured the local government, but never explicitly repealed the local legislation on segregation. SCOTUS ruled it had not been repealed, and that lack of enforcement by the executive branch does not a repeal make. As such the anti-segregation laws were still in effect and the result of the case was decided based on them.
- In Stoutenburgh v. Hennick (1889), the majority opinion written by Chief Justice Fuller said
It is a cardinal principle of our system of government that local affairs shall be managed by local authorities and general affairs by the central authority, and hence, while the rule is also fundamental that the power to make laws cannot be delegated, the creation of municipalities exercising local self-government has never been held to trench upon that rule. Such legislation is not regarded as a transfer of general legislative power, but rather as the grant of the authority to prescribe local regulations, according to immemorial practice, subject, of course, to the interposition of the superior in cases of necessity.
Congress has express power "to exercise exclusive legislation in all cases whatsoever" over the District of Columbia, thus possessing the combined powers of a general and of a state government in all cases where legislation is possible. But, as the repository of the legislative power of the United States, Congress, in creating the District of Columbia "a body corporate for municipal purposes," could only authorize it to exercise municipal powers, and this is all that Congress attempted to do.
In the parenthetical above these I linked an article, the topic of which is also relevant, as it concerns the implications of Immigration and Naturalization Service v. Chadha for the Home Rule Act. This case concerned a law which granted each chamber of Congress a sort of veto power over deportations. The ruling held that such single-chamber vetos were unconstitutional. The basis of the above article is that Congress can be argued to be exercising similar veto authorities via the Home Rule Act.
To my knowledge, Chadha or similar precedents have not been used in courts to challenge the Home Rule Act, and the rationale of Chief Justice Fuller quoted above is still good law to this day.