Grappling is for creatures; dragging and carrying is for objects
The fact that the grappling rules are meant to be used for creatures is fortunately clear from the rules themselves. Besides saying at the outset that they are to be used when you want to "grab a creature", the grapple is contested by the target's choice of a Strength (Athletics) or Dexterity (Acrobatics) check - and objects cannot choose between two abilities they do not have.
However, the rules for lifting and carrying do not make it explicit whether they are to be used for moving creatures, objects, or both. Instead, they simply refer to the weight you wish to move. A quick word search of my PHB finds that the word weight is used 48 times, and about equally between the weight of creatures (mostly characters) and the weight of objects (mostly equipment) - twice it is even used metaphorically to mean worth or importance. So weight is clearly not a game term, and thus interpreting any rules passage in which it occurs has to rely on context to determine natural use. Looking at the very next passage on the encumbrance variant though, one finds the interesting phrase (emphasis mine):
The rules for lifting and carrying are intentionally simple. Here is a variant if you are looking for more detailed rules for determining how a character is hindered by the weight of equipment.
There is thus some textual support for the idea that the lifting and carrying rules are meant to be used only for objects, not creatures.
Stat Blocks contain size, not weight
More convincing than the weak textual evidence, for me, is a purely pragmatic approach. Monster stat blocks list size; they do not list the weight of creatures. The grappling rules are modified, twice, by the size of the creature grappled; "The target of your grapple must be no more than one size larger than you", and "you can drag or carry the Grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you." For any creature a player would like to grapple, the DM knows the size class from its Stat Block and can thus adjudicate the grapple. This assumes the creature is a typical weight for its size class.
If the DM were expected to also apply the lifting and carrying rules to a grapple, however, they would need to know its weight, information that is critically missing from nearly every creature's description. It is difficult to believe that this is the intent of the lifting and carrying rules, or further that a living creature struggling against your grapple can be treated as fundamentally the same as the 'dead weight' of an object.
I find it far more practical to conclude that the grappling rules apply when you are moving creatures against their will, and the lifting and carrying rules apply when you are moving objects which do not resist.
But what if 'Heavyweight' is very heavy?
Some creatures may be much heavier than their size alone would indicate - for example, a Steel Predator. In this case, a DM should simply modify the grappling rules as they see fit - disallowing the grapple entirely, imposing disadvantage on the attempt, or further restricting the move that the grappler can make. The alternative - stopping the combat to calculate the exact weight of the creature, and then applying it what is left of a character's lifting and carrying capacity after deducting their current equipment - is simply too impractical to seriously consider.
If the DM has permitted the grapple but has already determined that in this case the 'Heavyweight' is "heavier than Grappler's carrying capacity, but not so heavy as to exceed Grappler's push/drag/lift capacity", then a restriction to moving not more than 5 feet a round is entirely appropriate. Positing this case as OP does, however, ignores what was involved in the DM making this decision to begin with - that they knew the weight of the creature as well as the surplus carrying capacity of the character. Those were not arrived at trivially and are unlikely to be immediately available in any given combat.