In the United States, the Senate exists to give small states more influence and counteract the greater population of the larger states. It is constitutionally enshrined in that role. At the time that the Senate was made directly elected, there was little support for reducing that role, particularly among small states. And remember, constitutional amendments require two thirds of the Senators to agree (or a constitutional convention, but that has never happened since the founding).
In the United Kingdom, the House of Lords was left over from a time when the lords had more power. The general trend has been to reduce that power. When support for reduction reached a sufficient level, it was done. Power switched from the House of Lords to the Commons.
In both cases, the change from unelected to elected would seem to be something of a side effect. First, the Senate was appointed by the democratically elected state legislatures. There were accusations of corruption in that process. Direct elections were considered less corrupt rather than more democratic. Second, the Lords were the legacy branch and the Commons the newer branch. Transferring power from the legacy to the newer branch is natural. It's merely an extension of the initial process.
The House of Lords did not exist because the Commons needed balanced. It would be more accurate to say that the Commons came into existence to balance the power of the lords.
The US solution, making Senators directly elected, would have made little sense in the UK. Rather than a natural change, it would have essentially eliminated the House of Lords and added a new House. But how would such a House have worked? There doesn't seem to be a movement in the UK to balance large population areas with small population areas. So what they would have had would have been essentially a second House of Commons, which would have been redundant.
TL;DR: while it is accurate to say that both switched from unelected to elected, the reasons behind the switches were quite different.