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In this review, we discuss and extend the study of the inclusive production of vector quarkonia, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>/</mo><mi>ψ</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="normal">Υ</mi></semantics></math></inline-formula>, emitted with large transverse momenta and rapidities at the LHC. We adopt the novel ZCW19<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow></mrow><mo>+</mo></msup></semantics></math></inline-formula> determination of fragmentation functions to depict the quarkonium production mechanism at the next-to-leading level of perturbative QCD. This approach is based on the nonrelativistic QCD formalism well adapted to describe the formation of a quarkonium state from the collinear fragmentation of a gluon or a constituent heavy quark at the lowest energy scale. We rely upon the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>NLL</mi><mo>/</mo><msup><mi>NLO</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> hybrid high-energy and collinear factorization for differential cross-sections, where the collinear formalism is enhanced by the BFKL resummation of next-to-leading energy logarithms arising in the <i>t</i>-channel. We employ the method to analyze the behavior of the rapidity distributions for double-inclusive vector quarkonium and inclusive vector quarkonium plus jet emissions. We discover that the natural stability of the high-energy series, previously seen in observables sensitive to the emission of hadrons with heavy flavor detected in the rapidity acceptance of LHC barrel calorimeters, becomes even more manifest when these particles are tagged in forward regions covered by endcaps. Our findings present the important message that vector quarkonia at the LHC via hybrid factorization offer a unique chance to perform precision studies of high-energy QCD, as well as an intriguing opportunity to shed light on the quarkonium production puzzle.