Acids used in Friedel-Crafts alkylation I am learning adding alkyl groups to benzene using Fridel-Crafts alkylation:

I know that it is very common to use $\ce{AlCl3/AlBr3}$ as Lewis acid catalyst in this reaction. But I am just wondering whether I can use a Bronsted acid, e.g., $\ce{H2SO4}$, as the catalyst or not. For example, in a $\mathrm{E1}$ or hydration reaction, $\ce{H2SO4}$ is added to alkene or alcohol, respectively, forming a carbocation, which is the same ultimate result when $\ce{AlCl3}$ is used (make the alkyl group more electrophilic).
I am thinking $\ce{AlCl3}$ is the only way if I am adding primary alkyl halide since in  $\mathrm{E1}$/hydration reactions, an actual primary carbocation is impossible to be formed (too unstable), while it exists in a complex with $\ce{AlCl3}$ (not entirely a carbocation, so can be formed).
SO, what is the problem with secondary/tertiary alcohols if I use the Bronstad acid? If there is a problem, way? Both Lewis and Bronsted acid ways will lead to carbocation rearrangements (I have not learnt the acylation).
Brønsted Acid has been used in Friedel-Crafts alkylation before (for example, Ref.1). According to this reference, when toluene use as the solvent, the F-C alkylation reaction of toluene is fruitful for tertiary and secondary alkyl bromides, tosylates, and alkenes (which give both carbocations upon protonation). However, the only primary bromide, which worked in this alkylation are benzyl and cinnamyl bromides. Other active primary bromides such as allyl and crotonyl bromides gave ditolyl products (e.g., crotyl group substituted first, then underwent double bond protonation followed by hydride shift to give secondary benzyl carbocation, which then reacted with the second toluene nucleus to give 1,1-ditolylbutane). Keep in mind that these reactions worked only with activated phenyl nucleus. For example, chlorobenzene gave only trace of products even with higher temperatures.
Later, this reaction is successfully used in Tandem-Friedel-Crafts alkylation reactions to synthesize hitherto unknown perylene derivatives (Ref.2):

References:

Mathew P. D. Mahindaratne, Kandatege Wimalasena, &quot;Detailed Characterization of p-Toluenesulfonic Acid Monohydrate as a Convenient, Recoverable, Safe, and Selective Catalyst for Alkylation of the Aromatic Nucleus,&quot; J. Org. Chem. 1998, 63(9), 2858–2866 (https://doi.org/10.1021/jo971832r).
Mark A. Penick, Mathew P. D. Mahindaratne, Robert D. Gutierrez, Terrill D. Smith, Edward R. T. Tiekink, George R. Negrete, &quot;Tandem Friedel–Crafts Annulation to Novel Perylene Analogues,&quot; J. Org. Chem. 2008, 73(16), 6378–6381 (https://doi.org/10.1021/jo800558c)(PDF).