Since the Home Office is a government agency which operates on a fixed budget it is unlikely they can just hire more employees as the number of applications rises. They can request a budget increase for the following year so they can hire more employees then, but the decision to approve the request is a political one outside of their hands.
Home Office delays are a deliberate political choice...
Home Office visa requests all require the payment of substantial fees, certainly enough to pay for the staff to process them. So you would have thought staffing levels would increase in line with applications without any recourse to annual fixed budgets. (Actually from memory, over the last few years, pax numbers are up 20%+ and staff numbers down 15%...)
Similarly for Heathrow etc not having enough immigration officers to process incoming arrivals... I estimate the average passenger costs less than £2 to process [1], while outgoing flights pay Air Passenger Duty at rates starting from £13 per passenger (economy/short haul) and £78 (economy/long haul). More than enough to pay for higher staffing levels.
[1] £2 is my quick estimate, based on 130m passenger movements in 2016, Border Force having 7670 staff in 2016-17, with annual FTE costs around £32k for base salary, NI, and associated variable employment costs.
Just because an agency charges a fee which covers or exceeds their costs doesn't mean they get to keep those fees. They may go to the government's general revenue fund or they may be used within the Home Office to fund other activities within the agency. Even if the fees are directly returned to the immigration agencies with the Home Office they still may not be authorized to hire additional staff without prior approval through the budgeting process. Most governments don't like to give agencies unfettered ability to increase agency staff.
I have no idea what the situation is with the Home Office. They very well may have the required authority to increase staff but to assume they do doesn't seem logical since this is against the norm for government agencies.
