Why HCl is used in Lucas test?
The use of concentrated H2SO4 is not amenable. Per a source:

Concentrated sulphuric acid produces messy results. Not only is it an acid, but it is also a strong oxidising agent. It oxidises some of the alcohol to carbon dioxide and at the same time is reduced itself to sulphur dioxide. ... It also reacts with the alcohol to produce a mass of carbon.

On the other hand, per Wikipedia, to quote:


  The Lucas test in alcohols is a test to differentiate between primary, secondary, and tertiary alcohols. It is based on the difference in reactivity of the three classes of alcohols with hydrogen halides via an SN1 reaction:[3]...The differing reactivity reflects the differing ease of formation of the corresponding carbocations. Tertiary carbocations are far more stable than secondary carbocations, and primary carbocations are the least stable.


And further:


  An equimolar mixture of ZnCl2 and HCl is the reagent. The alcohol is protonated by this mixture, and H2O group attached to carbon is replaced by the nucleophile Cl−, which is present in excess. Tertiary alcohols react immediately with Lucas reagent as evidenced by turbidity owing to the low solubility of the organic chloride in the aqueous mixture. Secondary alcohols react within five or so minutes (depending on their solubility). Primary alcohols do not react appreciably with Lucas reagent at room temperature.[3] Hence, the time taken for turbidity to appear is a measure of the reactivity of the class of alcohol, and this time difference is used to differentiate among the three classes of alcohols:


Now, with respect to HClO4, there is at least one catalytic study of HClO4-SiO2 during the reaction of benzyl alcohol (1 mmol) and thiophenol (1.1mmol) at room temperature under solvent-free conditions (see).