The price will only increase if there is demand. I think the existence of a single coin does not affect the price much. According to the simple rule of economics, the more buyers of a good, the more its price increases. Therefore, whether Bitcoin stands alone or not, it can proceed in a similar way.
There are many cryptocurrencies, but which one can you trust as much as Bitcoin? Your answer to this question will be similar to the answer you would give to the question asked in the topic.
in every market, competition is important to know the difference of their role. however, if there is no competition, i guess, people won't appreciate much about its existence.