Why use Avoir à instead of avoir besoin de?

Rainy, 

“ I hope I don’t have to wait too long … “ -  “ J'espère que je n'aurai pas trop longtemps à attendre “ . 

Stated like this, the statement is more neutral about the impact of waiting on him - he is only hoping ‘ not to have to do it’,  rather than additionally or alternatively expressing a ‘ need not to have to wait ‘, which could use ‘ avoir besoin de ‘. Neither is it describing an  obligation, a duty,  a want etc, 

French is more particular about separating these things that are more regularly conflated in English ( in my view ).  

See link to Laura Lawless site for French expressions with avoir

 

“ Avoir expressions

 

avoir à + infinitive    to have to do something
 
avoir besoin de   to need 

 

https://www.lawlessfrench.com/expressions/avoir-expressions/