Is having a 14-50 installed the cheapest way?
Not necessarily the least expensive. If you have to purchase the Tesla Mobile Connector then the 14-50 receptacle and Mobile Connector end up being more expensive than purchasing the Tesla Wall Connector.
Yes, when charging using a 240 Volt, 50 amp circuit up to 29 miles of range can be added per hour while charging using the Tesla Mobile Connector (limited to 32 amps), a bit more if you use the Tesla Wall Connector (would be enable charging at 40 amps on a 50 amp circuit that is hard wired (no 14-50 receptacle and plug.)
You should find out from an electrician if there is additional capacity in the electrical service panel for your condo to add a 50 amp circuit. There are lower amperage 240V circuits including 20, 30, 40 that can also enable you to charge at more than 4X or more your current charging speed.
20 amp, up to 14 miles added per hour
30 amp, up to 21 miles added per hour
40 amp, up to 29 miles added per hour
50 amp, up to 36 miles added per hour
60 amp, up to 42 miles added per hour
Assuming you drive 30 miles per day, on average what most people drive, any of these would be able to add 30 miles in approximately two hours or less when charging. If the electrician tells you that you can only add a maximum of a 30 amp circuit that would enable you to add up to 21 miles per hour when charging.