Yes, the blower should come on regardless of refrigerant charge. The problem could be either the blower or the pushbutton unit. Some more troubleshooting is in order.

Holson's right about the evaporator. That was the straw that broke the camel's back on my long-gone 300E.

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Rick Miley
2014 Tesla Model S
2018 Tesla Model 3
2017 Nissan LEAF
Former MB: 99 E300, 86 190E 2.3, 87 300E, 80 240D, 82 204D Euro
Chain Elongation References