Bottom line, there’s a reg that requires it…
“ (3) Must hold at least a third-class medical certificate—
(i) When exercising the privileges of a private pilot certificate, recreational pilot certificate, or student pilot certificate, except when operating under the conditions and limitations set forth in § 61.113(i);”
Unless you’re a student seeking Sport Pilot privileges…
“ (1) A person must hold and possess either a medical certificate issued under part 67 of this chapter or a U.S. driver's license when—
(i) Exercising the privileges of a student pilot certificate while seeking sport pilot privileges in a light-sport aircraft other than a glider or balloon;”
At some point in the process of working towards a Private Pilot certificate, you have to have a medical certificate. At what point exactly one transitions to not being able to say, “I’m training for my Sport Pilot certificate,” is probably somewhat variable.