The UK has a fairly simple election system: you don't vote in many elections and generally not very many at once (maybe three or so at max if there's general elections and local elections happening at the same time). The US has many more elected positions that any given voter can be given. While it's easy to imagine that there's maybe one or two more ballots to fill in for e.g. the presidential election, in reality a voter on election day in the US may be voting in near 100 different elections for posts all the way from the local level to the national level. Marking, collecting, and counting these votes is an effort on another level from what the UK system deals with, and even filling in the ballot is not a trivial task (especially with the focus on being able to automatically count the votes).

There's a lot of scope for technology to improve this system, and a lot of improvements which can be made to the technology which is already there, though I do believe you either need a paper trail (which is less convenient and harder for an individual voter to verify their vote was counted, but a lot easier to explain how the system works and is hard to tamper with and preserve anonymity of the vote), or a clear open cryptographic voting system (which has more convenience and may be able to produce greater trust in the system, but it's much harder to explain how it works and preserving anonymity is much harder. 'Blockchain' contributes nothing to this BTW, it's more or less a completely different class of protocols).

The current situation in some US states where there's black-box electronic voting machines which just output a count at the end of the day and don't even have particularly good security to prevent someone who isn't the government or manufacturer from tampering with them is an awful state of affairs (though thankfully they do seem to be falling out of favor after some backlash).