I actually visit businesses for work (H&S) all types of business and have done for years, long before the referendum. That's not what I see. The types of business you describe couldn't get the staff before the referendum and still can't now. Example, there are a lot of food manufacturers near me, the HR manager told me they've done everything to try to recruit local staff in the factory, they could basically pick their own hours, nobody would do it. In her words, 'who grows up dreaming of working in a chicken factory'. The office staff are almost always all British.
Care homes I find are a mixture of British and non British staff, I haven't seen that balance change since Brexit. And they still all complain that they can't get the staff. I do remember I visited a care home right after the referendum though. The manager said she felt so bad for the eastern European care workers though. They had tried to make the vote a bit of a day out for the residents and they had all voted Leave, grumbling about 'too many foreigners' while these care workers wheeled them down to the polling station.
Of all they Plumbers/builders/etc I visit, in all my years doing this, I don't think I've met one that wasn't British (SW England) . My own experience of trying to hire tradesperson to do work, you can't find one for love nor money, they're all massively expensive and completely run off their feet. Again, they complain that nobody wants to do the job anymore. Same with car mechanics.
I also meet lots of businesses that have moved some or all operations to the EU. Example, I visited an importer of a small industrial product (I don't want to name the product as it would be outing). Product was manufactured in the USA and Mexico, shipped to UK. UK office did sales, warehousing and distribution. 85% of sales were in Europe and product was shipped on from here. Office staff were all British (apart from MD who was Spanish) warehouse staff 50/50 British/European. Going on national numbers, half the staff there probably voted Leave. Company has now moved to the Netherlands. Ironically, the European staff could keep their jobs and move with it, the British staff, not.
Anyway, as you said, we each have are own windows to the world, but I'm just not seeing these sunlit uplands care workers are now supposedly living in.