There is some truth in what you say, but as an argument I'd say it's rather weak: Net migration levels over the last 10 years have mainly held between 200K and 300K/year, so it's probably not hard to extrapolate for the next few years. Even without free movement, you'd still have to deal with emigration, which just like immigration, will be highly dependant on other countries' economies. So while I'd agree that free borders makes planning a bit harder, but it's really just an incremental issue on a process that's already very imprecise in any case.
Besides, the actual net migration level the last 10 years has amounted to between 0.3% and 0.5% of the UK population per year. That's really not a high proportionate increase. If I was running a business and I complained that I couldn't plan properly because my client base was increasing by 0.5% a year and that was too much for me to cope with, I think you'd probably conclude that I must be completely incompetent! Yet this rate is being suggested as a reason for the Government being unable to plan!?! (NB. There is a possible get-out in some cases: That 0.3% to 0.5% nationally undoubtedly includes larger changes in some local areas).
You're right that building houses and health centres etc. takes time, but it shouldn't be taking years and years to do so (with the one obvious exception of actually training doctors - you're correct that that process necessarily takes some years).
And a more serious point in many cases is: What planning? You're talking about immigration harming the Government's ability to plan, but I see precious little indication that the Government is giving much thought to even trying to systematically plan in many cases (with a few good exceptions - rail being one obvious one). To take housing as an example: Where are the Government estimates of how many houses are likely to be needed in 10 years time and where, with accompanying plans for how the Government are going to make sure those houses get built, along with accompanying infrastructure? What has the Government done? Well, far from planning for the future, it's just passed the Housing Act which actually forces local authorities and housing associations to sell their houses (a move that we know often simply results in those same houses being let privately at much higher rents after a few years), yet restricts the ability of local authorities to build replacement council houses: A move that appears inexplicable, other than as an ideologically-based attempt to limit social housing. I know my local authority are doing their utmost to get more affordable houses built, but feel the Government is hampering them at every step. Based on past experience, in a couple of years time, immigration will no doubt trotted out again as a scapegoat for the resultant increased lack of housing when the real problem appears to be that the Government is too busy pursuing free market ideology to actually sort out the housing shortage (although I'd agree that immigration does add to the pressure, so it is an extra factor).